set theory: techniques and applications cura§ao 1995 and barcelona 1996 conferences

Set Theory

Set TheoryTechniques and ApplicationsCuracao 1995 and Barcelona 1996 Conferences

edited by

Carlos Augusto Di PriscoInstituto Yenezolano de Investigaciones Cientificas,Caracas, Venezuela

Jean A. LarsonUniversity of Florida.Gainesville, Florida, U.S.A.

Joan BagariaUniversitat de Barcelona.Barcelona, Spain

and

A.R.D. MathiasUniversity of Wales,Aberystwyth, Wales, U.K.

Springer-Science+Business Media, B.V.

A C.I.P. Catalogue record for this book is available from the Library of Congress.

Printed on acid-free paper

All Rights Reserved© Springer Science+Business Media Dordrecht 1998Originally published by Kluwer Academic Publishers in 1998.Softcover reprint of the hardcover Ist edition 1998

No part of the materialprotectedby this copyrightnotice may be reproduced orutilizedin any fonn or by any means,electronicor mechanical,includingphotocopying, recordingor by any informationstorageandretrievalsystem, without writtenpermission from the copyrightowner.

ISBN 978-90-481-4978-0 ISBN 978-94-015-8988-8 (eBook)

DOI 10.1007/978-94-015-8988-8

TABLE OF CONTENTS

Preface

List of Participants

Articles

vii

ix

MAXIM R. BURKEForcing axioms . . . . . . . . . . . . . . . . 1

JAMES CUMMINGSLarge cardinal properties of small cardinals 23

CARL DARBY and RICHARD LAVERCountable length Ramsey games . . . . . . 41

OMAR DE LA CRUZ and CARLOS A. 01 PRISCOWeak forms of the axiom of choice and partitions of infinite sets 47

MARTIN GOLDSTERNA taste of proper forcing . . 71

PIOTR KOSZMIDERApplications of p-functions 83Models as side conditions . 99

JEAN A. LARSONAn ordinal partition from a scale 109A picaresque approach to set theory genealogy 127

A.R.D . MATHIASRecurrent points and hyperarithmetic sets. 157

E.C. MILNER and SAHARON SHELAHA tree-arrowing graph . . . . . . . . . . . . 175

WILLIAM J . MITCHELLA Hollow Shell: Covering Lemmas Without a Core 183

CARLOS H. MONTENEGROPartition properties for reals 199

ERNEST SCHIMMERLINGCombinatorial set theory and inner models 207

STEVO TODORCEVICDefinable ideals and gaps in their quotients 213

PREFACE

This volume grew out of a pair of conferences both focused on techniques andapplications. The first conference, Combinatorial Set Theory and its Consequences,was a Joint US-VenezuelaConference co-organized by Carlos A. Di Prisco and JeanA. Larson. It was held June 26-30,1995 at the Avila Beach Hotel in Curacao in theNetherlands Antilles. The second conference, Techniques and Applications of SetTheory was co-organized by Joan Bagaria and Adrian Mathias. It was held June10-14, 1996 at the Centre de Recerca Matematica (CRM) in Barcelona, Spain.

In his speech at the opening of the new CRM buildings, Professor Hiltonmade a contrast between vertical and horizontal research. Vertical research, inhis metaphor, is the development of a branch of mathematics within itself, whentechniques become stronger, but where there is a concomitant danger of the subject becoming less accessible to outsiders. Horizontal research is the building ofbridges between these vertically developed branches of mathematics.

In the last twenty years, the interior techniques of Set Theory have developedenormously: one may point to advances in the inner model program, the conceptof proper forcing, the axiom of determinacy, the set theory of the real line, and soon. There are growing signs that the time is ripe for set theorists to seek to buildlinks with other areas .

One of the aims of the Barcelona meeting was to have a chance to learn some ofthe techniques developed by Stevo Todorcevic, so two of the four short courses atthe Barcelona meeting were devoted to his work. The course given by Max Burkefocused on partition relations and forcing axioms, including several applicationsto Topology and Measure Theory. Piotr Koszmider's course expounded two of themost successful tools developed by Todorcevic, namely, p-functions, and forcingwith models as side conditions, and included some applications.

In their respective courses, Martin Goldstern gave an introduction to properforcing and James Cummings treated large-cardinal combinatorial properties ofsmall cardinals.

In this volume the reader will find expanded versions of the four courses. Webelieve it provides a very good and accessible introduction to these beautiful, albeittechnically difficult subjects, and we hope they will be put to use to solve manymathematical problems.

The conference in Curacao focused on combinatorial aspects of set theory. Oneof its aims was to promote cooperation between researchers in this area of settheory from different parts of the world. Another aim was to stimulate set theoretical research in Latin America. Ordinals appear in several papers . Carl Darbyand Richard Laver discuss games played on ordinals. Jean Larson uses short scalesto build interesting ordinal partitions. A.R.D. Mathias studies closure ordinals of

viii

iterations defined using an initial point and continuous function mapping a Polishspace into itself.

Partition theorems are another theme. Omar De la Cruz and Carlos A. DiPrisco discuss weak forms of the axiom of choice and partitions of infinite sets.Eric C. Milner and Saharon Shelah build a rich graph so that for every coloringof its edges with two colors, there is an induced large tree all of whose edgeshave the same color. Carlos Montenegro looks at a partition relation viewed as ageneralization of the Hales-Jewett Theorem.

Inner model theory has enhanced our understanding of possible cardinalitiesand produced useful tools. William J. Mitchell makes a start on a program toclassify the failure of the Weak Covering Lemma at a Woodin cardinal ; ErnestSchimmerling surveys a hierarchy of square principles extracted from core modeltheory.

Stevo Todorcevic uses a covering property of definable sets of reals to analysedefinable gaps in definable ideals.

There is also an historical article by Jean Larson examining the mathematicalgenealogy of the participants of the two conferences.

We thank Jorge Martinez, director of the the Caribbean Mathematics Foundation, for his work as liason in Curacao and the support of his foundation. Additionally we thank the National ScienceFoundation (NSF INT-9503676) and CONICITfor their financial support of the conference in Curacao,

We thank the CRM director, Manuel Castellet, as wellas the CRM staff, MariaJulia, Consol Roca and Joan Codina for their invaluable assistance. We also wantto thank the Generalitat de Catalunya (CmIT) and the Ministerio de Educaciony Ciencia for their financial support.

Finally, we thank Arlene Williams of the University of Florida for her help inpreparing this document.

LIST OF PARTICIPANTS

Participants are listed with affiliations. The initials of Barcelona and Curacaoare used as superscripts Band C to indicate the conference(s) attended. Individualswho have a second affiliation in parentheses were at the latter institution duringthe conference.

OFELIA T. ALASc, University of Sao Paulo

DAVID ANTONBUniversitat de Barcelona

H. R . ANTONIUSc, Anton de Kom University, Suriname

JOAN BAGARIAB,c, Universitat de Barcelona

JAMES E. BAUMGARTNERc, Dartmouth College

ROGER BOSCHB, Universidad de Oviedo

MAXIM R. BURKEB, University of Prince Edward Island

MARfA CARRASCO Universidad Simon Bolivar

JAMES CUMMINGSB, Carnegie Mellon University

PATRICK DEHORNOyB, Universite de Caen

OMAR DE LA CRUZc, University of Florida(Instituto Venezolano de Investigaciones Cientificas)

CARLOS A. DI PRISCOB,cInstituto Venezolano de Investigaciones Cientificas

RAIMON ELGUETA" , Universitat Politecnica de Catalunya

DAVID ESPERE", Universitat de Barcelona

MARTIN GOLDSTERN", Technische Universitat Wien

C. W . GORISSONc, Anton de Kom University, Suriname

VILLE HAKULINENB, University of Helsinki

JAMES M. HENLEc, Smith College

IGNASI JANEBUniversitat de Barcelona

ALEXANDER S. KECHRISc California Institute of Technology

MENACHEM KOJMANc, Ben Gurion Universityand Carnegie Mellon University

PIOTR KOSZMIDERBAuburn University

ADAM KRAWCZYKBWarsaw University

CLAUDE LAFLAMMEc, University of Calgary

JEAN A. LARSONc, University of Florida

x

RICHARD LAVERc , University of Colorado

JEFFREY LEANINGc , University of Florida

JIMENA LLOPISc, Universidad Simon Bolivar

J ORDI LOPEZB, Universitat Autonoma de Barcelona

MENACHEM MAGIDORc , Hebrew University

M. VICTORIA MARSHALLc , Universidad Catolica de Chile

A.R.D. MATHIASB, University of Wales at Aberyswyth,

(Centre de Recerca Matematica, Belleterra)

GISELA MENDEZc Universidad Central de Venezuela

ARNOLD W. MILLERc , University of Wisconsin, Madison

ERIC C. MILNERc , University of Calgary

WILLIAM J . MITCHELLc , University of Florida

CARLOS MONTENEGROc , Universidad de Los Andes, Bogota

CHARLES J .G . MORGANB, Merton College, Oxford

JUAN CARLOS MARTINEZB, Universitat de Barcelona

JUI?ITH ROITMANc University of Kansas

ERNEST SCHIMMERLINGc , University of California, Irvine(Massachusetts Institute of Technology)

ELiAS TAHHAN-BITTARc , Universidad de Los Andes, Trujillo

STEVO TODORCEVICc , University of Toronto

CARLOS E. UZCATEGuf, Universidad de Los Andes, MeridaPAULI VAISANENB

, University of Helsinki

ANDRES VILLAVECESc, Universidad Nacional de Colombia(University of Wisconsin, Madison)

W . HUGH WOODINc , University of California, Berkeley

PIOTR ZAKRZEWSKIB, Warsaw University

FORCING AXIOMS

MAXIM R. BURKElDepartment of MathematicsUniversity of Prince Edward IslandCharlottetown, P.E.I., NB C1A 4P3Canada

Abstract. This paper surveys combinatorial forcing axioms together with combinatorial and topological consequences and information about their consistency.

These notes provide a survey of some forcing axioms which are consequencesof the proper forcing axiom. The notes are a modified and expanded version ofthe author's lectures given at the workshop "Techniques and Applications of SetTheory" held in June 1996 at the CRM in Barcelona, Spain. We thank S. Todorcevic for extensive discussions of the material. We thank Maria Julia of the CRMfor transcribing the author's transparencies into TeX. The level of detail variesfrom section to section. Section 1.4 especially - in which we give a proof, basedon a technique in [5], of the preservation of the properness isomorphism conditionunder short countable support iterations - is much more detailed than the restof the paper. It is imported from an unpublished note of the author's. In the restof the paper, we have added to the content of the transparencies some references,a few proofs and some comments, but there are many details left to the reader .For unexplained set theoretic notation, see [6J or [12] .

1. A property of ideals

The first axiom, R(·) defined below, is a common generalization of several resultsof S. Todorcevic. It is a special case of Lemma 1* (p. 152) of [14] and is easily seento be equivalent to Theorem 6A of [4] (see Remark 1.1.4 below). The statement isa type of Ramsey theorem concerning certain types of partitions of the countablesubsets of a set S , hence the choice of the letter R.

1Research supported by NSERC. The author thanks the Centre de Recerca Matematica for financial support during his stay in Barcelona and the Department of Mathematics at the University of Wisconsin-Madison for its hospitality while this paper wasbeing written.

C.A. Di Prisco et al. (ed.), Set Theory, 1-21.© 1998 KluwerAcademic Publishers.

2 MAXIM R. BURKE

After introducing appropriate definitions, we will give several applications ofR( ·), followed by a proof that it follows from the Proper Forcing Axiom.

1.1. DEFINITIONS

Definition 1.1.1. Let I be an ideal of subsets of a set S containing all singletons.For the purposes of these notes, we will let R(I) denote the following statement:

R(I) : either: S =U{An In < w} where, for each n < W , [An]W ~ t,or: for every f : S X WI -t I there is an uncountable A ~ S

such that An f(x, 0:) is finite for every z E A, 0: < WI .

Let R(·) denote the statement that R(I) holds for every ideal I.Remark 1.1.2. If I is generated by N1 sets, then we can let {f(o:, x) I 0: E WI }

be a generating family of I for each xES to see that R(I) implies

either: S =U{An In < w} where, for each n < W, [AnJw ~ Ior: 3A E [S]W\ \IF E I (IA n FI < w)

Remark 1.1.3. If, for each x E S, f(o:, x) does not depend on 0:, then we maythink of f as function from S into I . Then by thinning out the set A so that thesets An f(x) for x E A, form a ~-system and by discarding the root we mayassume that the sets An f(x) for x E A, are pairwise disjoint . Consequently wemay assume that if x E f(x) for all xES, then (A n f(x) = {x}) for all x E A.Remark 1.1.4. The following equivalent formulation of R(·) is Theorem 6A of[4J. Let S ~ X x Y.

either: there is a sequence (Yn In < w) of subsets of Y such that Y \(U{ Yn I n < w} is countable and for every countable B ~ Yn

there is a finite I ~ X such that B ~ S[I]or: whenever T ~ X x Y has horizontal sections of size :5 N1 (and

h : Y -t X) there is an uncountable A ~ Y such that An S[{x}]is finite for every x E T-1[A] (and (h(y),z) rt S for all distinctY,z E A) .

We leave to the interested reader the easy exercise of verifying the equivalenceof the definition of R(·) with the formulation given in Remark 1.1.4.

Variations on the statements in Remarks 1.1.2 and 1.1.3 above are studied in[8]. It followsfrom the proof of Theorem 3.12 of [8] that in the presence of Martin'sAxiom, the conclusion in Remark 1.1.2 can be improved to say that one of thefollowing possibilities holds:

(a). S =U{An In < w }, where [An]W ~ I for all n < Wi

(b). S =U{An In < w}, where IAn n FI < w for all n < wand F E I; or(c). there are A, BE [S]W\ so that [A]W ~ I and IB n FI < w for all F E I .

1.2. SOME CONSEQUENCES OF R(·)

The following consequences of RO are taken from [10], [l1J, [12]. For pedagogicalreasons this list is far from minimal.

FORCING AXIOMS

Theorem 1.2.1. Assume R(·) . Then there are no Sousl in trees.

Sketch of proof. Let T be an wI-tree . Let I the ideal generated by the sets

pred(x) := {y E T IY ~T x} for x E T .

3

Every set in the ideal is a finite union of chains, since the generating sets are allchains. By R(I) with f(x) := pred(x) and Remark 1.1.3,

either: T = U{An In < w}, where [An]n ~ I for n < w, which impliesA is a countable union of chains;

or: there is an uncountable set A ~ T such that

Anpred(x) = {x}, x E A,

which implies A is an antichain.

Either alternative shows T is not a Souslin tree. 0

Theorem 1.2.2. Assume R(·) . Then b > WI '

Sketch of proof. Suppose b = WI ' Let F ~ WW be unbounded and ordered by <*in type WI . Choose 1 - 1 functions

ex : Fx -t W for x E F

such that for all x E F, if Fx := {y E Fly < x}, then

{ey rFx lyE F} is countable.

Define H : F -t P(F) by H(x) := {y E Fx =1ex(Y) ~ x(A(x, y))}, whereA(x, y) := min{ n I x(n) :I y(n) }. Note that H(x) is either finite or defines a"sequence" converging to x.

Let I be the ideal generated by the sets H(x) for x E F . By Remark 1.1.3 andR(I) with f = H,

either: F = U{Fn I n < w} where [Fn]W ~ I for each n < w, which isimpossible, since any uncountable Fn has a perfect set of accumulation points ;

or: there is an uncountable F ' ~ F such that

which we now show is also not possible.

Let D ~ F' be a countable dense set. Fix c E F' so that d <" c for all dE D.Let m < w, F" ~ F', e : Fe -t W, S E wm satisfy

(1). F" is uncountable;

(2). c(n) < x(n) for all n ~ m and all x E F";

(3). s ~ x and ex rFe =e, for all x E F";

4 MAXIM R. BURKE

(4). {x(m) Ix E F"} is infinite.

Pick any d E D such that 5 ~ d. Choose x E F" such that

x(m) > max{d(m), e(d)} .

Then Ll(x, d) =m, d <* x and ex(d) =e(d) < x(m), so d E H(x), a contradiction.o

Theorem 1.2.3. Assume R(·). Then every regular hereditarily separable spaceis Lindelof.

Sketch of proof. Let X be a regular hereditarily separable space and assume byway of contradiction that X is not LindelOf. Thus we can choose points X a E Xand open sets Va such that Xa E Va \ U{ V,8 1/3 < Q} for all Q < WI . Since{Xa I Q < WI} is not Lindelof, we may assume X = {xa I Q < WI }.

Since X is regular, we may assume that V a ~ {X,8 I /3 ~ Q} for all Q< WI '

Let I be the ideal generated by the sets Va, for Q < WI and apply Remark 1.1.3and R(I) with f(xa) := Va to see that

either: X = Ux; In < w, where [Xn)W ~ I for n < w, which is impossible since [XnjW ~ I implies that Dn E I for any countabledense D« ~ x; and hence that x; ~ {X,8 I /3 < Q} for someQ <WI;

or: X has an uncountable subspace A such that A nVa = {xa} forall X a E A, which implies A is discrete, contradicting hereditaryseparability. '

o

Theorem 1.2.4. Assume R(·). Then every partial order with no uncountable weakantichains is a countable union of directed sets .

Sketch of proof. Let E be a partial order with no uncountable weak antichains.Let I be the ideal generated by the sets

pred(x) := {y EEl y ~ x} for x E E.

The second alternative in R(I) produces an uncountable weak antichain in E, andhence we have

For each i there is an ni ~ 1 for which there are

Si ~ [E)W, stationary, and

F; E [Ejni for all s E s,

such that for every s E Si and every x E B, n 5, there is ayE F; so that x ~ y.Write F; := {f;(j) 11 s j s nil·

FORCING AXIOMS 5

We may assume that the sets B, are disjoint and that for every i < w andevery j with 1 ~ j ~ ni, there is a k(i,j) < w so that

For each i < w, fix an ultrafilter Ui on S, extending the club filter on Si . For eacha E B, there exists ja E {I, . . . ,nil such that

{ 8 E s, Ia s /;(ja) } E u;

We have Bi = U{B{II ~ j ~ nil where B{ := {a E B, I ja = j}. Choosel(i,j) E {I, . . . ,nk(i,i)} such that

{ S I f i ( 0) Bl(i,i)} U8 E i s J E k(i ,i) E i ·

The setsDi - Bi U Bl(i,i) U Bl(k(i,i),l(i,m U

i - i k(i ,i) k(k(i,i) ,l(i,m· . •

are directed andE =U{ D{ I i < w, j =1, .. . ,nil .

o

Next turn to the classification of cofinal types of directed sets of size Nl .

Definition 1.2.5. For directed sets Db D 2, write D, ~ D 2 if there is a mapsp : Dl -t D2 such that <p[S] is unbounded for all unbounded S ~ Di,

o, and D2 are cofinally similar (D l == D 2) if o, ~ D 2 and D 2 ~ o;Theorem 1.2.6. Assume R(·). Then every directed set of cardinality aleplv; iscofinally similar to one of

The diagmm below indicates the relationship of the five types to each other. (In thediagmm, a and b are joined by a line segment, with a to the left of b, to indicatethat a ~ b.)

w

1 / ~wxw~/l

Wl

Sketch of proof. Suppose D is a directed set of cardinality Nl . The first part of thediagram is easy to sort out, so we must show

or

6 MAXIM R. BURKE

Let I be the ideal generated by the sets pred(x) := {y ED I y ~ x} for xED(i.e., the ideal of bounded sets) .

By R(I), we have

either: D = Un<w ti; [Dn]W ~ o,or: there is an uncountable set A ~ D such that IA n FI < W for all

FEI.

If the first alternative holds, then let cp : D -t W X WI be any 1 - 1 functionsuch that cp[Dn ] ~ {n} x WI, and notice that cp witnesses that D ~ W X WI. If thesecond alternative holds, then any 1 - 1 function cp : [wd<w -t A ~ D witnesses[wd<w ~ D. 0

Next consider partition relations of the form WI -t (WI, a)2.

Remark.

1. WI -t (Wl'W + 1)2 is a theorem of ZFC.

2. b = WI implies WI -It (WI, W + 2)2 .

3. MA + c =W2 implies c -It (c,w + 2)2.

Proof. For the first part, see [3]; for the third part, see [7]. To see that the secondpart holds, let F ~ WW be unbounded and ordered by <" in type WI as in theproof of Theorem 1.2.2. Using the notation of that proof, define a partition [FF :=Ko U K 1 into disjoint sets where for each x <* yin F, {x,y} E K 1 if and only ifx E H(y). Note that H(x) n H(y) is finite whenever x :I y, and recall from theproof of Theorem 1.2.2, that in every uncountable F' ~ F there are x <* y suchthat x E H(y). 0

Theorem 1.2.7. Assume R( ·). If p > WI, then Wl-t (WI, a)2 for all a < WI .

Sketch of proof. Suppose [wlF = Ko U K 1, and assume there are no uncountableO-homogeneous sets.

We show by induction on a < WI that for every uncountable A ~ WI, there isa set C ~ A of order type wOt and an uncountable set B ~ A above C such that

a =0: Clear.a > 0 : write wa = Ln<w wOtn where ao ~ al ~ .. . < a.Let I be the ideal of subsets of A generated by the sets K t (a) = K 1(a) n A

for a < WI, where K 1(a) = {13 < a I {13, a} E K 1 } . Since the mapping a ~Kt(a) U {a} has no uncountable free sets, by R(I) , A = U{ An In < w}, where[An]W ~ I for n < w.

One of the An's is uncountable .

For notational simplicity assume [A]W ~ I and for each ~ < WI choose F~ E[Wl]<w such that An~ ~ U{K1(71) 171 E F~} .

Let E ~ WI be uncountable such that {F~ I ~ E E} is a 6-system, and bythinning out the set A, assume the root is empty.

FORCING AXIOMS 7

Using the induction hypothesis, inductively choose sets Gn , Bn satisfying (*)so that

Go, Bo ~ A, Gn+! , Bn+! ~ s; and otp(Gn ) = wOn .

Then G' =U{ c; In < w} has type WO and [G']2 ~ tc;Since p > Wi, there is an ultrafilter F on G' such that if 1£ ~ F has size:::; Ni

then there is a Gil ~ G' with otp(Gil) = o"; such that for all H E 1£, there issome ~ E Gil so that Gil \ ~ ~ H. [Exercise [Hint: proceed by induction on 0:], orsee [10].]

By thinning out the index set E, we may assume Fe is above G' for all ~ E E,and E is above G'. For each ~ E E, fix Ye E Fe such that G'n K1 (Ye) E F . Get Gilas above and find a fixed ~o E Gil and uncountable Eo ~ E such that for all ~ E Eo,Gil \ ~o ~ G' n Ki(Ye) ~ Ki(ye) . The sets G = Gil \ ~o, and B = {Ye I ~ E Eo}are as desired. 0

1.3. PFA AND R( ·)

Theorem 1.3.1. PFA implies R( ·).

Proal. Let I be an ideal of subsets of a set 8 containing all singletons. Let i be thea -ideal generated by the sets A ~ 8 such that [A]W~ I. Assume that lis proper.Fix I :8 x Wi -t I . We must find an uncountable set A ~ 8 such that A n I (x, 0:)is finite for all x E A, 0: < Wi. Fix a well-ordering <w of 8 in type 181. Let P bethe set of all conditions p = (Ap,Bp,Np) such that

(a). Ap E [8]<w, Bp E [wd<w j

(b) . Np is a finite E-chain of countable elementary submodels of He where

(J = Ill+j and

(c). Ap is (Np,I)-separated in the sense that

(i) if a<w b are in A p then 3N E s; aEN, b rt N;

(ii) Va E Ap "'IN E Np art N => art U(ln N).

Order P by p :::; q if and only if Ap ;2 Aq, N p ;2 Nq, Bp ;2 Bq, and (Ap \ Aq) nl(a,{3) = 0 for all a E Aq and {3 E B q •

Claim. P is proper.

Proof. Proof of claim Fix PEP, a regular /'i, much larger than (J, M < HK. countablesuch that p, P, . . . EM.

Let q = (Ap,Bp,NpU {M n He}). We will show that q is (M,P)-generic.Let DE M be a dense open subset and r :::; q. We may assume rED.Let f := (Ar n M, B; n M,Nr n M), m = IAr \ Arl.Let F:= {F E [8]m =13s :::; f sED and send-extends f and As \ Ar = F}Note that FE M n He.To prove the claim, we need the following fact .

8 MAXIM R. BURKE

Fact. If J is a a-ideal on 8, F ~ [8]m, J,FE Mo E M1 E . .. E Mm-1 a chain ofcountable elementary submodels of Hk, FE F is J-separated by {Mo, .. . ,Mm-tl,then there is F* ~ F, F* E Mo such that F* i' 0 and "IE if E is an initial partof a member of F* then

{x E 8 lEU {x} is an initial part of a member of:F* } rt J.

Proof. Exercise.[Hint: If F = {ao, .. . ,am-tl with ai rt U(J n Mi), i < m, and ai E Mi+l fori < m-1, then Tm - 1 = {x E 8 Iam - 2 <w x and {ao, . . . ,am - 2,x } E F} E Mm - 1

and Tm - 1 contains am-I, SO Tm - 1 rt J. Now notice that Tm - 2 = {x E 8 Iam-3 <w x and {y E 8 I x <w yand {ao , . .. ,am-3,x,y} E F} rt J} E Mm - 2and Tm - 2 contains am - 2 , so Tm - 2 rt J. Continue this backwards induction.] 0

By the fact, with J = I, F = Ar \ Ai'", M n Hs = Mo, . . . , Mm- 1 from Nr, getF* EM. Since 80 ={x E 8 I {x} is an initial part of a member of F* } rt I, thereis some Do E [80 ]Wso that Do E M and Do rt I.

Pick Xo E Do \ (U{ f(a, (3) I a EAr, (3 E s, }).Let 81 = {x E 8 I {xo,x} is an initial part of a member of F*}. Then 81 rt I,

so there is some D1 E [81]Wso that D1 EM and D1 rt I .Pick Xl E D1 \ (U{ f(a,(3) I a E Ar (3 E n, }).Continue, getting

{xo, . . . ,xm-tl E F* n M.

By def of F, there is s $ f sED so that s end extends f, As \ Ai'" ={xo, ... ,xm-d, and we may take s E M . It is easily seen that s and r are compatible.

This completes the proof of the claim that P is proper. 0

The set A = U{Ap I pEG} for a sufficiently generic G has the desiredproperties. 0

Note. If we fix a family 11. ~ P(8) of cardinality N1 and demand in the conclusionof R(I) that An H is countable for every H E 11. and strengthen the hypothesis tosay that the o-ideal generated by IU1I. is proper, then the proof above shows thatthis new statement also follows from FA. This stronger statement is Lemma 1* of[13].

1.4. THE PROPERNESS ISOMORPHISM CONDITION

In the next section, we sketch a consistency proof, without using large cardinals, fora restricted version of R(·) . The technique is based on the material in [9] startingon page 262 and in [12] . We begin by giving, in this section, a proof of the requiredpreservation theorem from [9] .Lemma 1.4.1. If M, N are extensional sets and hI, ba: M -+ N are E-isomorphisms, then hI = ha-

FORCING AXIOMS 9

Proof. By E-induction on x E M, we show hl(x) = h2(x) .

Let v E hI (x)nN. Write v = hI (u). Then u E x, so by the induction hypothesis,we have hl(u) = h2(u) . We also have h2(u) E h2(x) . Thus v = hl(u) = h2(u) Eh2 (x) and hence hI (x) n N £; ba (x) . By this inclusion and the one obtained byinterchanging the indices 1 and 2, we get hl(x) n N = h2(x) n N. Since N isextensional, it follows that hl(x) = h2(x). 0

Lemma 1.4.2. Let M and N be countable elementary submodels of H8 for someuncountable regular cardinal O. If MEN, then M and N are not E-isomorphic.

Proof. Suppose 1r: M ~ N were an isomorphism. Since MEN and M is countable, we have M £; N. Define a sequence ~ M n : n < w) of elements of N asfollows: Mo = M and for all n < w, 1r(Mn+d =Mn.[This makes sense since, by induction, Mn E N so there is Mn+1 EM£; N suchthat 1r(Mn+d = Mn.]

By induction on n, we get Mn+I E Mn for all n, contradicting the axiom offoundation.

[MI E M = Mo, and if Mn+I E Mn, then 1r(Mn+2 ) = Mn+I E Mn = 1r(Mn+I)and hence Mn+2 E Mn+I since 1r is an E-isomorphism.] 0

Lemma 1.4.3. If M and N are elementary submodels of H8 for some regularuncountable cardinal 0, n : M ~ N is an E-isomorphism and A = {x E M I1r(x) = x} is the set of fixed points of 1r, then

(a) . A £; M n N.

(b). [A]Wn M £; A, i.e., countable sets of fixed points are fixed.

(c). 1r r(M n HW 1 ) = id .

(d). MnHw\ = NnHw\ '

Proof. Part (a) is obvious; (c) follows from (b) by showing by E-induction onx E M n Hw\ that 1r(x) = Xj (d) follows by applying (c) to both 1r and 1r-I • Thereremains to prove (b) .

Let x E [A]w . In M, pick a surjection f: w ~ x. Since M 1= Vy(y E x)if and only if there is some n so that y = f(n)), we have N 1= Vy(y E 1r(x)if and only iff there is some n so that y = 1r(J)(n)). For any n < w, if we lety = f(n), then we have y E z. So by the induction hypothesis, 1r(Y) = y. Thus1r(J)(n) = 1r(Y) = Y = f(n). Hence, 1r(J) = f and it follows from the equivalencesabove that 1r(x) = X. 0

Lemma 1.4.4. If P is a partial order, N is an elementary submodel of H8 forsome regular uncountable cardinal 0, <8 is a well-ordering of H8, and G £; P is ageneric filter over V , then

(N[G], E, <~ rN[G],H8n N[G]) -< (He[G] , E, <~,H8)

where the fourth coordinates are one-place predicates and <~ is the well-orderingof Be[G] defined as follows: for x, y E Be[G], we write x <~ y when a <8 T where

10 MAXIM R. BURKE

a is the <o-least P -name in Ho such that x = aa and T is the <o-least P-namein Ho such that y = rc -

Proof. It is clear that <~ is a well-ordering. The rest of the proof is essentially thesame as the proof that (N[G], E) -< (Ho[G], E). We use Tarski's criterion. Let T bea vector of P-names in N and suppose that for some formula ¢ in the language ofthe structure (Ho[G] , E, . . . ), we have

(Ho[G], E, ... ) 1= 3x¢(x,Ta) .

Now replace each occurrence of <~ in ¢ by its definition in terms of <0 and bythe maximal lemma find a P-name a E Ho such that

(Ho ,E,<o) 1= Irp "3x¢(x,T) ~ ¢(a,T)" .

[We have to make sense in (Ho, E, <0) out of expressions like p Ira E Ho andp Ira <0 b. This can be done using standard tricks. For example, interpret theseexpressions as meaning Vq ~ p 3r ~ q3x (r Ira = x) and Vq ~ p 3r ~ q 3x 3y (x <0y and r Ira = x, b= y), respectively.]

By elementarity, a can be chosen in N and then we have (Ho[G] , . . . ) 1= ¢(aa, Ta)and oo E N[G] . 0

Definition 1.4.5. Let '" be a regular cardinal larger than Wl ' A partial orderP satisfies the '" properness isomorphism condition, or more succinctly, is s-pic,if and only if for all large enough regular cardinals (), the following condition issatisfied:

If a < {3 < n, a E No, {3 E N{3, «, P E No n N{3, No and N{3 are countableelementary submodels of the structure (HIJ , E, <IJ) for some well ordering<0 of Ho, No n '" ~ {3, No n a = N{3 n {3,

pEP n No and 11" : No ~ N{3 is an E-isomorphism such that 1I"(a) = 11"({3)and 11" r(No n N(3) = id,

then there is a (P, No)-generic condition q extending both p and 1I"(p) such that

qlr 1I""(GnNo) = GnN{3.

Remark 1.4.6. Let P be a partial order and let (),a,{3,No,N{3,p,1I" be as in thehypothesis of the statement in the definition of the x-pic.

(a). For any ordinal ~ E No n N{3 n «, we have ~ E No n '" ~ {3 and so~ E N{3 n {3 = No n a . Thus ~ < a < {3 < '" and No n ~ =N{3 n~.

(b) . Since Wl, P E No n N{3, we have 1I"(wI) =Wl and 1I"(P) = P . (Of coursefor Wl this also follows from absoluteness of Wl for submodels of Ho.)

(c). No n HW 1 = N{3 n HW 1 by Lemma 1.4.3.

Remark 1.4.7.

(a). If there is a A < '" for which AW ~ n, then every partial order is vacuously x-pic. More precisely, there do not exist (),a, {3, No, N{3, 11" as in thehypothesis of the statement in the definition of the x-pic ,

FORCING AXIOMS 11

(b). It is not clear whether the x-picis an isomorphism invariant. If, in thedefinition of the x-pic, we add after the phrase "for all large enoughregular cardinals B" the phrase "there exists an x E He such that" , andadd to the hypothesis on the models that x E Na n N{3 , then the x-picbecomes an isomorphism invariant and nothing essential changes in thetheory which follows.

To see part (a), suppose such objects existed. The least A < I\: for whichAW ~ I\: is in both models, and so is the <e-Ieast 1-1 function g : I\: -t [A]w .We have Na n A = N{3 n A by Remark 1.4.6(a) , and so 1l" rNa n A = id and hences; n [A]W = N{3 n [A]W by Lemma 1.4.3(b). Thus Na n I\: = N{3 n 1\:, contradicting1l"(a) ={3.

With the modified definition of part (b), if PI is x-pic and P2 is isomorphic toPI, then given B, if x E He witnesses that PI is x-pic, then {x,Pd witnesses thatP2 is x-pic by the argument used in the proof of the next proposition.

Proposition 1.4.8. Any proper partial order of cardinality less than I\: is «-pic.

Proof. Let P be a partial order of cardinality A < 1\:. Let B be a regular cardinallarger than 1\:, and let a,{3,Na,N{3 ,p,1l" be as in the hypothesis of the statement inthe definition of the x-pic for P . Then A and the <e-Ieast bijection g : A -t P belongto Na nN{3. Since A E Na nN/3 n1\:, by Remark 1.4.6(a) we have Na nA = N/3 nA.Thus Nan P = N{3nP is contained in Nan N(3 and 1l" maps this set identically toitself. The condition q required by the conclusion of the statement in the definitionof the x-pic is now given by the properness of P . 0

Example 1.4.9. If A < I\: then Fn(A, 2,wd is «-p ic. This follows from the previous proposition if AW < 1\:. Otherwise it is vacuous by Remark 1.4.7(a).

Lemma 1.4.10. Let P be a s-pic partial order. Let a, {3, B, Na, N{3 ,1l",p be as inthe hypothesis of the statement defining the s-pic. If q is a (P, Na)-generic condition given by the conclusion of the statement, then

(a) q is also (P,N/3)-generic.

(b) q It- "1l" extends to an E-isomorphism 1T : Na[G]-t N{3[G] given byIT(aa) =1l"(a)a where a ranges over P -names in Ni,".

(c) q It- 1T r(Na[G] n N/3[G]) = id .

Proof. (a) If D ~ P is dense, D E N/3, then 1l"-I(D) E Na is also dense in P.Let G be any generic filter containing q. Then G n 1l"-I(D) n N« :j:. 0 and hence(G n N/3) n D = 1l""(G n Na) n D :j:. 0. Thus N/3 n Dis predense below q.

(b) The formula for 1T is unambiguous because if ac = ra for some P-namesa, r E N , then there is apE Na[G] such that p It-a = r . Since q is (P, Na)generic, q It-Na[G] n V = Na and hence p E Na. Thus 1l"(p) 1t-1l"(a) = 1l"(r) . Wehave 1l"(p) E 1l""(G n Na) = G n N/3 ~ G and thus 1l"(a)a = 1l"(r)a .

12 MAXIM R. BURKE

For x E N«, it(xa) =1I"(x)a =1I"(X)a = 1I"(x) so that it is indeed an extensionof 11".

As in the proof that it is well-defined, we can show that aa E Ta impliesit(ua) E it(Ta) for P-names U,T E Na. Since 11"-1 is also an isomorphism , it

extends to a function from Np[G] into Na [G] by letting ;;:=t(aa) = ;;:=t (a)a forP-names a E Np. Also, this extension preserves the membership relation. Sincethe extensions of 11" and 11"-1 are clearly inverses of each other, the extension of 11"is an isomorphism, as desired.

(c) If x E Na[G]nNp[G], let a be the <~-least P-name in H() such that x =ua .by Lemma 1.4.4 above, we have a E Na[G] n Np[G] and hence, since q is (P, Na)generic and (P,Np)-generic, a E Na n Np. Hence it(x) = it(ua) =1I"(u)a =Ua =x. 0

Lemma 1.4.11. If K, is a regular cardinal larger than WI, PI is K,-pic and 11- Pl

"K, is regular> WI and P2 is «-pic", then PI *P2 is n-pic.

[By corollary 1.4.13(a,b) below, I~PI "K, is regular> WI" is automatic if AW < K,

for all A < K" and this is the only case of interest by Remark 1.4.7(a).]

Proof. Let f) be a regular cardinal which is "large enough" for the purposes of thedefinition of the x-pic for PI and such that I~PI "f) is large enough for the purposes

of the definition of the x-pic for P2". Choose a,/3, Na,Np, (PI,'h) E PI *P2, 11" beas in the hypothesis of the statement in the definition of the K,-pic for PI*P2• ThenPI E Nan Np and PI E Na SO there is a (PI, Na)-generic condition ql $ PI,1I"(Pdsuch that

ql 1~1I""(01 n Na) = 0 1n Np.

(01 is the canonical PI-name for the generic filter added by forcing with PdIf Gl is generic, qi E Gl, then in V[Gll we have a < /3 < K" a E Na ·~ Na[Gl]

and (Na[Gd, E, . .. ) -< (H()[Gd, E, ... ) (see Lemma 1.4.4). Similarly for /3 andNp[Gl]. Also, Na n Np contains a PI-name equivalent to P2. (Such a PI-namecan be manufactured using T = {a2 I (aI, a2) E PI *Pd and {(P, a,b) I p EPI, a, bET and p I~a s b}.) Hence P2 [Gl] E Na[Gl] n Np[Gd. Since ql is(PI, Na)-generic, we have Na[GdnV =Na and similarly with /3 in the place of a.Thus Na[GdnK, ~ /3 and Na[Gdna =Np[Gdn/3 since these relations hold withoutthe [GIl'S. We have also PI,1I"(Pd E Gl and, as shown in Lemma 1.4.10(b,c), wehave an extension of 11", which we shall denote also by 11", to an isomorphism betweenNa[Gl] and Np[Gd fixing their intersection. We also have 'h[Gd E Na[Gd.

Thus, there is an (Na[Gl],P2 [Gd )-generic condition q2 $ 'h[Gd,1I"(P2[Gl])such that

q21~1I""(Na[Gl]n (2) = Np[Gd n 02.

Let q2 be a PI-name for q2 and let q = (ql,Q2). Then q is (Na,Pl *P2)-genericand there remains to show that q I~ 1I""(Na n G) =Np n G. It will suffice to showq I~ 1I""(Na n G) ~ Np n G, the reverse inclusion being similar .

FORCING AXIOMS 13

Let u,a be such that U= (Ul,U2) :$ q = (ql,Q2) and UII- a = (al,a2) E NcrnG.Then Ul :$ ql and Ul II- al E Gl n Ncr and so Ul 11-1I"(ad E Gl n Nf3 . Moreover,

Ul II- "U2 :$ Q2 and u211-a2 E Ncr [Gl] n G2",soulll-u211-1I"(a2) E Nf3[GI]nG2 and hence U= (Ul,U2) 1f-1I"(a) = (1I"(al),1I"(a2)) EN{3 n (Gl *G2) = Nf3 n G. Thus q11-1I""(G n Ncr) ~ G n N{3, as desired . 0

Lemma 1.4.12. If f'i, is a regular uncountable cardinal such that >,w < f'i, for all >. <f'i" and (J ~ f'i, is any regular cardinal, then for any countable expansion (Ho, E, . . . )of Ho and any x E Ho, there is a set S ~ f'i, of cardinality f'i, and there are countableelementary submodels Ncr of (Ho, E, . .. ) for 0 E S such that {Ncr I 0 E S} isa 6.-system and for any distinct 0,(3 E S , we have o,x E Ncr, Ncr and N{3 areE-isomorphic and if 11" : Ncr ~ N{3 is the unique isomorphism then 11"(0) = (3 and11" r(Ncr n N{3) =id. Furthermore, if 0 < (3 then Ncr n f'i, ~ (3 and Ncr no =Nf3 n (3.

Proof. For each 0 < f'i" choose Ncr -< (Ho, E, . .. ) such that Ncr is countable ando,x E Ncr. By the pressing down lemma, there is a stationary set S ~ {o < f'i, Icf'(o) = WI} such that for some 'Y < f'i, and all 0 E S we have Ncr n 0 ~ 'Y. Byassumption, there are less than f'i, countable subsets of 'Y and hence we can thinout the set S so that for 0 E S, Ncr n 0 does not depend on o . After a bit morethinning out, this takes care of the "furthermore" part of the statement. Thinningout further still , we can arrange that the set { Ncr I 0 E S} is a 6.-system with rootR, say. The number of isomorphism types of the structures (Ncr, E, 0, c (c E R))(0 and the c's are constants) is at most c < f'i, and hence by thinning out S somemore we may assume that for 0 E S these structures are pairwise isomorphic . 0

Corollary 1.4.13. If f'i, is a regular uncountable cardinal such that >.w < f'i, for all>. < f'i" and P is n-pic then

(a) Every countable set in vP is covered by a countable set in V . In particular, P preserves WI •

(b) P is n-ee and even satisfies property KK, '

(c) For any>. < f'i" vP1= >.W < f'i,. In particular, vP 1= c < n ,

Proof. (a) Identical to the proof of the corresponding fact for proper partial orders.The lemma guarantees the existence of models for which the hypothesis in thestatement of the definition of the x-pic applies.

(b) If :$ Pcr : 0 < f'i,) is a sequence of members of P, apply the Lemma with(J > f'i, and x = {P,:$ Pcr 10< f'i,) } to get Ncr, 0 E S. We have 0 E Ncr and hencePcr E Ncr for each 0 E S . For 0 < (3 in S, since 11"(0) =(3, we have 1I"(pcr) =P{3 andhence (see the definition of the x-pic) Pcr and 1I"(pcr) = P{3 are compatible.

(c) This can be shown directly as follows. Use (a) and (b) to see that P collapsesneither WI nor f'i" then fix pEP and use Lemma 1.4.12 to get 0 < (3 < f'i" Ncr, N{3, 11"etc . Then take any generic filter containing a generic condition q :$ p given by thedefinition of the x-pic and notice that, as in the proof of Lemma 1.4.10, Ncr[G]and Nf3[G] are such that by Remark 1.4.7(a) we must have vP 1= >.W < f'i, for anyx< n,

14 MAXIM R. BURKE

The same result can be obtained indirectly using the following argument from[2]. We can assume A ~ WI. Let Q be a P-name for Fn(A,2,wd. By parts (a)and (b), I~p "K, is a regular cardinal> WI". By example 1.4.9, I~p "Q is x-pic"and hence, by Lemma 1.4.11, P *Q is x-pic, Thus P *Q is K,-CC by part (b) . Inparticular, P *Q preserves K,. Since Q collapses AW to A, we have I~ p AW < K,. D

Example 1.4.14. Fn(c", 2) is c++-pic but is not c+-pic. The failure of c+-picfollows from corollary 1.4.13(c). The c++-pic follows from proposition 1.4.8.

Theorem 1.4.15. Let K, be a regular cardinal larger than WI. Let (~ p{ : ~ ~

c), ~ Q{ : ~ < c)) be a countable support iteration of partial orders such that e < K,

and for all ~ < e, I~{ UK, is regular> WI and Q{ is «-pic", Then Po is «-pic.

[It then follows by induction on ~ ~ e, using corollary 1.4.13, that the assumptionI~{ "K, is regular> WI" holds automatically if AW < K, for all A < K" and 1~1j "Q1jis x-pic" for all n < ~.]

Proof. Let 0 be a regular cardinal such that for all ~ < e, I~{ "0 is large enough for

the purposes of the definition of the x-pic for Q{". Choose 0, f3,NOt, N{3,p E Po,1ras in the hypothesis of the statement in the definition of the x-pic for Po. .

As in [5], the proof will proceed by showing, by induction on TJ, that for all~ < TJ , if ~, TJ E NOt n (s + 1), PE NOt is a P{-name for a condition in P1j , and

(a). q E p{

(b). q is (p{ , NOt)-generic

(c). q Ih 'Pt~, 1r(PH~ E Ge(d). q Ih 1r"(G{ n NOt) = G{ n N{3,

then there is a condition q+ such that

(a"). q+ E P1j, q+ f~ = q

(b+). q+ is (Pe,NOt)-generic

(c"). q+ 1~1j 'PtTJ, 1r(p) fTJ E G1j

(d+). q+ 1~1j 1r"(G1j n NOt) = G1j n N{3.

[The desired result then follows by taking ~ = 0, TJ = s, p =p.]

The initial step TJ =0 is trivial and the successor step is essentially the same asthe proof of Lemma 1.4.11. [By the induction hypothesis, we may assume ~ = TJ -1.]

For the limit step of the induction, write U(TJ n NOt) = Un<w'Yn, 'Yn E NOt,'Yo = ~ < 'Yl < 'Y2 < ... . Let {Dn I n < w} enumerate the dense subsets of P'1which are in NOt.

Exactly as in 3.18 of [5], define {Pn I n < w} so that Po = Pand for all n < W ,

Pn E NOt and

(a) Pn is a Porn -name for a condition in P1j

FORCING AXIOMS 15

(b) IL • <.,- -Yn+1 Pn+l _ Pn

(c) If-", PnH E DnIn+l

(d) If--Yn+1 "If Pn f"YnH E G-Yn+1 then PnH f"YnH E G-Yn +1

Now define (qn : n < w) so that qo = q ~ ql ~ ... , and for all n < W , qn+l f"Yn = qnand qn satisfies (a), (b), (c), (d) with (q,p,~) replaced by (qn,Pn,"Yn).

[qnH = q; is obtained using the induction hypothesis (with (~,1l), P replaced by("Yn ,"Yn+d, Pn f"YnH respectively) . By (c'"),

By (3), qnHIf--Yn+1 PnH f"Yn+I E G"ln+1 ' By (d"), qn+I/f--Yn+1 7r1/(G"ln+1 nNo.) =G-Yn+1 n Nfl and hence qnH If-"In+1 7r(PnH f"Yn+d = 7r(Pn+d f"Yn+1 E G-Yn+1')

Let q+ = Un<w qn' We must check (a+)-(d+).

(a"): clear.

(b"), (c"): Let G l1 be generic, q+ E G l1• Let p., = Pn[G-Yn)' Since qn is (No.,P-yJgeneric, we have Pn = Pn[G-y,.] E No.[G-y,.] E No.[G-Yn) n V = No.. Since q+ E G l1

and q+ ~ qn, we have qn E G l1 and hence Pn f"Yn E G-Yn'

By (1), for any k ~ n we have Pn S; Pn-l S; ... ~ Pk· Thus Pk f"Yn E G-Yn forall n ~ k and hence Pk f8 E Go where 8 = sUPn "Yn by 1.17 of (5) . Since Pk E No.,supp(pr) ~ No.ne ~ 8 and hence pi, =pf8 E Go ~ Gl1 and we have pi, E c.r.t«;Since Pk f"Yn E G-Yn for n ~ k and Pk f"Yn E No. and 7r1/(G-Yn n No.) = G-Yn n Nfl,we have 7r(pd f"Yn E G-Yn [by (d) with qn,"Yn in the place of q,~ using 7r("Yn) = "Yn) .Thus 7r(pd E G l1 by 1.17 of (5). In particular, for k = 0, we have 7r(p) E G11' Thisestablishes (c" ). Also, PHI E o, n Gl1 n No. [see (2)). This proves (b+).

(d+) follows from q+ S; qn 1f-7r I/ (G-Yn n No.) = a.; n Nfl and 1.17 of [5]. 0

1.5. R{·) FOR IDEALS ON WI WITH WI GENERATORS

We will now sketch a proof very similar to the proof that PFA implies R(·) toshow that

Con{ZFC) implies Con{ZFC + R{·) for ideals on WI with WI generators)

We can ask also ask for Martin's axiom or even PFA for partial orders of cardinality WI in the conclusion. We start with a model of CH and perform an w2-stagecountable support iteration of w2-pic forcing notions. By the general theory presented in the previous section, the initial segments of the iteration are w2-pic, inparticular they are W2-CC. It follows that the final iteration is W2-CC and preservesWI. We assume that the reader is familiar with the type of bookkeeping argumentneeded to build such a model and simply explain how to handle a single stage ofthe iteration.

Fix an ideal I of subsets of WI containing all singletons and generated by{So. I a < WI }. Assume that the a-ideal J generated by the sets A ~ WI such that

16 MAXIM R. BURKE

[A]W ~ I is proper. We must force an uncountable subset of WI which has finiteintersection with each BQ •

Let M be the set of all countable elementary submodels N of HW2 such that(BQ : a < WI) EN. For two isomorphic NI,N2 EM, let 1rN1N2: N I ~ N2 denotethe unique isomorphism. Let us write N I ~ N 2 to mean that N I and N 2 areisomorphic and 1rN1N2 t(NI n N2 ) = id. Let Mo = {F E [M]<w I F i- 0, F is a~-system and VNI,N2 E F, N I ~ N2 } . For FE Mo, let 8(F) denote the commonvalue of N nWI for N E F. On Mo, define FI EF2 to mean (a) VNI E FI 3N2 E F2

N I E N2 and (b) VN,N' E F2 VW E FI (W E N implies 1rNN'(W) E Fd.Define a partial order P to be the set of all conditions p = (Ap,Bp,Np) where

Ap and Bp are finite subsets of WI, Np is a finite E-chain, for all a, b E Ap if a < bthen a < 8(F) ::; b for some F E Np, and for any a E Ap and F E Np if a ;::: 8(F)then a ¢ U(J n UF). The order on P is given by q s p iff Aq ~ Ap , s, ~ e;"IF E Np3F' E Nq F ~ F', and "1/3 E s, (Aq \ Ap)n B{3 = 0.

We have to show that P is w2-pic and forces the desired set. Most of this is verysimilar to the corresponding parts of the proof that PFA implies R(·), so we onlysketch the argument. Let (J be a large regular cardinal, and let a, /3, NQ, N{3, 1r =1rNaNp be as in the hypothesis of the statement in the definition of the w2-pic.If p = (Ap,Bp,Np) E NQ then the desired generic condition q ::; p is given byq = (Ap,Bp,:N"puHNQnHw2 ,N{3nHw2}}) whereN;, = {FU(1rNaNp)"F: F E Np}.We leave to the reader the verification that this condition has all the desiredproperties.

2. The open coloring axiom

Much of the material in this section has been well covered elsewhere, so we willlimit ourselves mostly to listing results . See [12] and also the survey article [16] forthe omitted proofs.

Definition 2.1. For a Hausdorff space X, let OCA(X) be the following statement:

OCA(X): If [X]2 = Ko UK I is a partition with Ko open (in the exponentialtopology) then either there exists an uncountable O-homogeneousset, or X is a countable union of l-homogeneous sets.

Theorem 2.2. OCA(wW) holds.

Proof. Let [wWF = Ko U K I, Ko open. Suppose WW is not the union of countablymany I-homogeneous sets .

Let

X =WW \ U{U : Uopen, U is the union of countably many

I-homogeneous sets} .

Then X is closed and non void and no nonvoid open subset of X is a countableunion of l-homogeneous sets.

Fix a complete metric for X . Let U0 = X and inductively define clopen setsUs, s E 2n, 1 ::; n < W such that

FORCING AXIOMS

(a) Us ¥ 0;

(b) s ~ t => Us ;2 Uti

(c) u.:; n Us~1 =0i

(d) diam(Us ) ~ ~j and

(e) {{x,y} I x E Us~o,y E Us~l} ~ tc;00

Let Y = nuUs .n=lsE2n

Then Y is a O-homogeneous Cantor set .

Theorem 2.3. If OCA(X) and f: X -t Y is a continuous surjection,then OCA(Y).

Corollary 2.4. For every analytic space X, OCA(X) holds.

Proof. Suppose [Yj2 = Ko U K 1, Ko open.Define

Lo = {{a, b} E [X12 I f(a) ¥ f(b) and {f(a) , f(b)} E Ko }

and

17

o

L 1 = [X1 2\ Lo.

Then [xj2 =Lo U L1 and Lo is open.If A ~ X is Lo-homogeneous then 1"A ~ Y is Ko-homogeneous . Also, f is

1 - 1 on A and hence 1"A is uncountable if A is uncountable (and 1"A is a Cantorset if A is a Cantor set).

If X = U{x, I n < w} , where [Xn12 ~ L1 for n < w, then Y = 1"X =U{1"Xn I n < w}, and for any a,b E Xn we have either f(a) = f(b) or{f(a) , f(b)} E K 1 • Thus [1"xnF ~ K 1 •

This establishes OCA(Y). 0

Remark 2.5. OCA(wW) becomes false if "open" is replaced by "closed".

Remark 2.6. PFA implies OCA(X) for all X ~ JR.

Let OCA abbreviate "OCA(X) for all X ~ JR."

Remark 2.7. OCA implies

(a) for all X E [IR1W\ and all f : X -t IR,there is Y E [X1 W

\ so that fry is monotonic.

(b) b =W2i

(c) Every automorphism of P(w)jfin is trivial.

Problem 1. Does OCA => C = W2? (PFA => C = W2)

18 MAXIM R. BURKE

Problem 2. (See [1].) Does OCA =} th ere are no strong homomorphisms h : A -+B where

B = Borel a-algebra of (0,1).A = subalgebra of B generated by sets of the form00

U(ai, bi), ai < bi < a2 < bz < ... orn=i00

U(bi , ai), bi > ai > bz > a2 < ...,n=iand h is strong means:

1. I ~ h(I) for all open intervals I ~ (0,1) .2. The finite subsets of (0,1) are all in Ker(h).

3. A set mapping principle

The results in this section are taken from [15] .

Definition 3.1. 8Mo is the following statement: for every F : 0 -+ [O]:Sw suchthat V~ E 0 (~ f/. F(~)),

either: 0 = UAn, An F-free (i.e., V~ E An (F(~) nAn = 0))n<w

or: there is an uncountable A ~ 0 such that for every finite r ~ A,{ TJ < 0 I r ~ F(TJ) } is uncountable

M AB=} 8MB =} Every tree on 0 which has no uncountable chain is special.

Theorem 3.2. M AB =} 8MB continues to hold after forcing with a measure algebra.

Theorem 3.3. 8Mw l =} there are no compact L-spaces.

Proof of 3.3. Suppose X were a compact L-space.Let Y = {Yo I a < Wi} be a left-separated subspace, Yo E U0, U0 open,

Uo n Yo =0, Yo = {y~ I~ < a} .Define F : Wi -+ [wd:Sw by F(a) := {~ < a IYo E U~ }.The first alternative in 8Mw! produces an uncountable discrete subspace. The

second alternative produces an uncountable set A ~ Wi'

Let C ~ Wi be a club such that for each 0 E C, 0 = N nWi for some countableN --( HI< such that Y, A, (UO)O<WIl FEN.

Then for each 0 E C ,{ U~ n Yo I~ E An0}

is centered, so choose Xo E n U( n Yo . Since X is compact, { xo I 0 E C} has~EAno

a complete accumulation point p.Since X is 1st countable, we can find 0 <~, 0 E C, ~ E A, such that

P E {xo' I0' E C no}.

FORCING AXIOMS

But then, p E Yo and for each d' E C

d' > ~ :::} Xo' E U{ .

So, since p is a c.a.p .,

P E {xo' Id' > ~, d' E C } ~ U{ .

Thus Yo n U{ ;f; 0, contradiction.

Proof of 3.2 (Sketch).

19

D

We start with a pair of lemmas.

Lemma 3.4. The following are equivalent:

(1) Every measure algebra has precaliber NI , i.e., every uncountable subsetof a measure algebra R has an uncountable centered subset.

(2) For every sequence ([f{J : ~ < WI) of members of some ultrapower R1 jUsuch that V~ < WI limuJ.LUdi)) > 0, there is an uncountable 0 ~ WI and

t-ta sequence C{, ~ E 0 , of elements of R \ {O} such that L I1 f{(i) ~

i EK { Er

I1 C{ ;f; 0 for all K E U, r E [OJ<w.{Er

Proof of (I):::} (2): Force with R. In V[GJ define a finitely additive measure m :P(I) -t [0, 1J as follows:

(K) I· I' J.L([i E K] . a)m =Imlm .a-tG H U J.L(a)

We may assume

3£ > 0 V~ < WI lim J.L(/{(i)) > e .t-tU

Define K{, ~ < WI, by[i E Kd = /{(i) .

Show 3ao E R forcing

is uncountable.Thus ao II- B = P(I)jNn is an algebra with a strictly positive finitely additive

measure and uncountably many of the [K{] are non-zero in B.By Lemma 3.5 below, B has precaliber NI , so there is an uncountable A ~ WI

such that m( nK{) > 0 for all r E [AJ<w.{Ef .

In the ground model find an uncountable 0 ~ WI such that ~ E 0 impliesC{ = [~ E A] > O.

By (1) we can assume {C{ I ~ EO} is centered. These C{ are as desired. D

20 MAXIM R. BURKE

Lemma 3.5. If every measure algebra has precaliber NI then this remains trueafter forcing with a measure algebra.

Proof. Exercise. oBack to the proof of Theorem 3.2. Let R be a measure algebra, 0 an uncountablecardinal.

Assume MAe. Let F be a name for a set mapping .We may assume If- F(a) ~ a , all a < O.The sets

~(a) = {~ < a I [~ E F(a)] t: O}

are countable.Let Ra be a separable non atomic subalgebra of R containing

{ [~ E F(a)] I~ E ~(a) } .

Let P be the set of all functions p such that domp E [O]<w and

(a) p(a) ERa and j.t(p(a)) > t for all a E domp.(b) p(a) . p({3) ~ [a ~ F({3)] for all a < (3 in dom(p).

For p,q E P, let P ~ q if domp 2 domq and p(a) ~ q(a) for a E domq .The generic object is an R-name 0 for an F-free subset of 0 such that p(a) ~

[a E OJ for every p and a E domp.If P is ccc, MAe applied to P" gives the first alternative of 8Me .

Claim A. If P is not ccc, then some condition in R forces the second alternativeof 8Me.

Proof. Let Pe, ~ < WI, be allegedly pairwise incompatible elements of P. By thinning out this collection, we may assume that (a) {domp- I ~ < WI} is a .6-systemwith root D, (b) for some n < wand all ~ < WI we have Idom(Pe) \ DI = n,(c) for some e > 0, j.t(pe(a)) > 1/2 + e for all a E dompe and ~ < WI, (d)~(a)n(dom(Pe)\D) = 0forall ~ < TJ and a E dompe, and (e) j.t(Pe(a)·Pl7(a)) > 1/2for all ~,TJ < WI and a ED. Because of (e), we may as well assume that D = 0.For each ~ < WI, let (c4 :j < n) be the increasing enumeration of domPe and, for

i ,j < n, let g~j E RWl be given by g~j(TJ) = [ak E F(a~)] for TJ > ~ and g~j(TJ) = 1otherwise. Fix a uniform ultrafilter U on WI and find i, j < n and an uncountableno ~ WI such that for all ~ E 0 0 , {TJ < WI I j.t(g~j (TJ)) ~ e/n2

} E U. (This requiressome thought.) By Lemma 3.4, there is an uncountable nI ~ 0 0 and a sequence(ce : ~ E ( 1) of elements of R \ {O} such that

L II g~j(TJ) ~ II cel7>~eEr eEr

for all 'Y < WI and r E [OI]<w , Choose a condition c in R \ {O} which forces thatthe set O2 of ~ E nI for which ce belongs to the generic filter is uncountable. It isnow straightforward to verify that c forces that A = {a~ I ~ E 02 } satisfies thesecond alternative of 8Me . 0

FORCING AXIOMS

The above claim completes the proof of Theorem 3.2.

References

21

o

1. Burke M.R., Liftings for Lebesgue measure, in: Set theory of the reals (Proceedingsof a winter institute on set theory of the reals held at Bar-Ilea University, RamatGan (Israel), January 1991), (H. Judah, ed .), Israel Math. Conf. Proc. 6 (1993)119-150.

2. Dow A., On the consistency of the Moore-Mrowka solution, Topology Appl., 44(1992) 125-142.

3. Erdos P., Rado R. , A partition calculus in set theory, Bull . Amer. Math. Soc., 62(1956) 427-489.

4. Fremlin D.H., Consequences of Martin's Maximum, Note of July 31, 1986.5. Goldstern M., Tools for your forcing construction, in: Set theory of the reals (Pro

ceedings of a winter institute on set theory of the reals held at Bar-Ilan University,Ramat-Gan (Israel), January 1991), (H. Judah, ed.) Israel Math. Conf, Proc., 6(1993) 305-360.

6. Jech T ., Set Theory, Academic Press, New York, 1978.7. Laver R. , Partition relations for uncountable cardinals :£ 2No , in: Infinite and finite

sets, (A. Hajnal, R. Rado, and V. S6s, eds.), Colloq. Math. Soc. Janos Bolyai 10vol. II, 1975.

8. Nyikos P., Piatkiewicz L., On the equivalence of certain consequences of the properforcing axiom, J. Symbolic Logic, 60 (1995) 431-443.

9. Shelah S., Proper forcing, Springer-Verlag, Berlin, 1982.10. Todorcevic S., Forcing positive partition relations, TI-ans. Amer. Math. Soc., 280

(1983) 703-720.11. Todorcevic S., A note on the proper forcing axiom, in: Axiomatic set theory,

(J.E. Baumgartner, D.A. Martin, and S. Shelah, eds.) Contemp. Math. 31, Ameri can Mathematical Society, 1984, 209-218.

12. Todorcevic S., Directed sets and cofinal types, 1hms. Amer. Math. Soc. 290 (1985)711-723.

13. Todorcevic S., Partition problems in topology, Contemp. Math. 84, American Mathematical Society, 1989.

14. Todorcevic S., Some applications of Sand L combinatorics, Ann. New York Acad.Sci. 705 (1993) 130-167.

15. Todorcevic S., Random set mappings and separability of compacta, Topology Appl74 (1996) 265-274.

16. Velickovic B., Applications of the open coloring axiom, in: Set theory of the continuum, (H. Judah, W. Just and W.H. Woodin, eds), Math. Sci. Res. Inst. Publ, 26,Springer-Verlag, New York, (1992) 137-154.

LARGE CARDINAL PROPERTIES OF SMALL CARDINALS

JAMES CUMMINGS1

Department of Mathemat ical SciencesCarnegie Mellon UniversityPittsburgh PA 15213USA

Abstract. The fact that small cardinals (for example N1 and N2 ) can consistentlyhave properties similar to those of large cardinals (for example measurable orsupercompact cardinals) is a recurring theme in set theory. In these notes I discussthree examples of this phenomenon ; stationary reflection, saturated ideals and thetree property.

None of the results discussed here is due to me unless I say so explicitly.

1. Large cardinals and elementary embeddings

We begin by reviewing the formulation of large cardinal properties in terms ofelementary embeddings. See [40] , [22] or [21] for more on this topic.

We will write "j : V ---t M " as a shorthand for the rather cumbrous assertion"M is transitive, j and M are classes of V and j is a non-trivial elementaryembedding from V to M ".

If j : V ---t M then it is easy to see that j has a critical point K. That is tosay j r K = id; and j(K) > K . It turns out that many large cardinal propertiescan profitably be formulated in terms of elementary embeddings and their criticalpoints.

The concept of a measurable cardinal was first considered by Ulam [42] inconnection with problems in measure theory. Scott [35] initiated the study ofelementary embedding formulations for large cardinals by proving

Theorem 1.1 (Scott [35]). The following are equivalent.

1. K is measurable (that is, there exists a normal measure on K).2. There exists j : V ---t M such that crit(j) = K.

3. There exists j: V ---t M such that crit(j) = K and K. M ~ M .

1I would like to express my thanks to Joan Bagariaand Adrian Mathias for organizinga very enjoyable meeting.

23

CA. Di Prisco et al. (ed.), Set Theory, 23-39.© 1998 Kluwer Academic Publishers.

24 JAMES CUMMINGS

Purists may worry about the quantification over proper classes in the statement "There exists j : V ~ M .. . " . These worries can be addressed either byregarding Theorem 1.1 as a theorem schema or by working in a theory which allowsquantification over classes.

Other large cardinal properties can be defined by demanding that the "targetmodel" M of the embedding should have some resemblance to V. Here are threepopular large cardinal properties defined in terms of elementary embeddings:

Definition 1.2. Let K. be a cardinal.

1. K. is ,x-strong iff there exists j : V ~ M such that crit(j) =K., ,x < j(K.) andVA~M.

2. K. is ,x-supercompact iff there exists j : V ~ M such that crit(j) = K.,

,x < j(K.) and AM ~ M.3. K. is huge iff there exists j : V ~ M such that crit(j) = K. and j(K) M ~ M .

It is worth noting that all of these large cardinal properties have equivalentdefinitions which do not involve elementary embeddings and just assert the existence of an appropriate set ; see [30] for the case of ",x-strong cardinal" and [40]for the cases of ",x-supercompact cardinal" and "huge cardinal" .

By Theorem 1.1 only cardinals at least as strong as a measurable cardinal canhave this kind of definition as the critical point of j : V ~ M . However weaklycompact cardinals can also be defined using a weaker form of embedding, and thiswill be useful later.

Fact 1.3 (Keisler). K. is weakly compact iff K. is strongly inaccessible and forevery transitive M such that K. EM, <KM ~ M, IMI = K. and M models enoughset theory there exists k : M ~ N an elementary embedding into some transitiveset N with crit(k) = K..

Hauser [19J has given similar formulations of many properties intermediatebetween weak compactness and measurability.

One advantage of formulating the large cardinal properties of a cardinal K. interms of elementary embeddings with critical point K. is that it tends to make the"reflection properties" of K. very clear. We illustrate with an example which willbe a paradigm for several later arguments.

Fact 1.4. Let K. be measurable, let 8 ~ K. be a stationary set . Then there exists aregular cardinal Q < K. such that 8 n Q is stationary in Q (we say 8 reflects at Q).

Proof. Fix j : V ~ M with crit(j) = K.. Now j(8) n K. = 8, so 8 E M. Thestatements "K. is regular" and "8 is stationary" are downwards absolute (as theyare expressed by III sentences). Also K. < j(K.) because K. = crit(j). Hence

M l= "K. is regular and j(8) n K. is stationary and K. < j(K.)" .

By the elementarity of j ,

V l= "there is regular Q < K. such that 8 n Q is stationary" .

o

SMALL CARDINALS 25

In fact the assumption that K. is weakly compact would suffice to prove Fact1.4. The proof is very similar to the one we just gave, the key point being that forevery S we may build an appropriate M with S E M and then apply Fact 1.3.

In what follows we will be concerned with a more general kind of elementaryembedding. We'll write "k : M ~ N" to abbreviate "M, N are inner models ofZFC and k is a non-trivial elementary embedding from M to N" . In this generalsetting we are not assuming that k and N are classes of M or even that N ~ M.

It's worthwhile to bear in mind the following differences between the specialcase j: V ~ M and the general case k: M ~ N.

- If j : V ~ M with crit(j) = K. then

• K. is measurable.

• V,,;+l ~ M .

• V f:. M.

• j t V,,; = idv, .- In the general case there can be k: M ~ N where (at one extreme) M = N ,

or (at the other extreme) where crit(k) = N~ and Vw"-r-l S;; V~l'

We'll be particularly interested in the case of embeddings j : V ~ M ~ V[GJwhere i ,M are defined in V[G], a generic extension of V . These are usually knownas generic elementary embeddings; Foreman initiated the detailed study of genericembeddings and their applications in [9].

It will be convenient for us to assume that V-generic filters exist; it is possibleto eliminate this assumption, using any of the standard methods. Our forcingterminology followsthat of [25] for the most part. We write Add(K.,>') for the Cohenconditions to add>. subsets of K., Coll(K., >.) for the Levy conditions to collapse >.to have cardinality K., and Coll(K., < >.) for the Levy conditions to collapse eachordinal less than >. to have cardinality K..

To build generic embeddings we will use the following basic result of Silver.

Fact 1.5 (Silver). Let k : M ~ N, let IP' E M be a forcing poset. Suppose thatGis lP'-generic over M, H is k(IP') -generic over Nand k"G ~ H. Then there existsa unique k* : M[G] ~ N[H] such that k" t M = k and k*(G) = H .

Proof. If such a k* exists then it must map rO to k(r)H for each lP'-term rEM.We need to check that this gives a well-defined elementary embedding.

Suppose that rO = aO. Then there is pEG such that p II-r a = r, so byelementarity k(p) If-~p) k(a) = k(r). Now k(P) E k"G ~ H, so that k(a)H =k(r)H and we have proved that k* is well-defined. The proof of elementarity isvery similar . 0

We list some ways to arrange that k"G ~ H will hold. Fix k : M ~ N andIP'EM .

1. If If ~ k(If), k t If =id p, and G =H n If then clearly k "G ~ H.

26 JAMES CUMMINGS

2. Suppose M 1= "lP is < A-distributive" and N = { k(F)(aF) IF E :F} where:F S;;; M is a family offunctions such that 'tIF E :F M 1= Idom(F) 1< A.Then we claim that k"G generates a filter H which is k(JP)-generic over N.To see this let DEN be dense in k(lP), then D = k(F)(aF) where withoutloss of generality F(x) is dense in lP for each x E dom(F). By distributivityE = nXEdom(F) F(x) is dense in P, and of course E E M, so let pEG n E;then by elementarity k(P) E k(F)(aF) =D and so k"G n D =10.

3. If q E k(lP) and 'tip E G q ~ k(P) then any k(lP)-generic filter H such thatH 3 q will also have the property that H ;2 k"G. Silver dubbed such acondition a master condition.

2. Stationary reflection

Recall that in the last section we proved that every stationary subset of a measurable (or even weakly compact) cardinal reflects. We now consider the possibilitiesfor this kind of phenomenon in small cardinals like N2 and Nw+l'

We introduce some convenient terminology for describing stationary sets.

Definition 2.1. S; ={a < AIcf(a) = J.L } . T~ ={a < Nm Icf(a) =Nn l -It is easy to see that full stationary reflection cannot hold at the successor of

a regular cardinal A. In fact st is a stationary subset of V, but if a < A+ thencf(a) ~ A so we can choose C club in a such that C n st = 0.

On the other hand it is consistent that stationary subsets of TJ should allreflect. More precisely Baumgartner [2] proved the following

Theorem 2.2 (Baumgartner [2]). If K. is weakly compact and G is generic overV for the Levy collapse Coll(N1 , < 11:) then

V[G] 1= "If S ~ TJ is stationary, there is a E T[ with S n a stationary".

Proof. For simplicity we assume that K. is measurable (and will indicate at the endof the proof how to weaken the assumption to weak compactness).

Fix j : V ~ M such that crit(j) = K. and KM S;;; M . Let lP =Coll(N 1 , < 11:).Then by the closure of M we have j(lP) = Coll(Nl, < j(II:)), so in the naturalway j(lP') := lP x Q where Q = Coll(N 1 , [K.,j(K.))) . If G is lP'-generic over V andH is Q-generic over V[G] then G * H is j(lP)-generic over V and so a fortiori isj (lP')-generic over M .

What is more, for every pEG we have j (P) =pEG *H, because G S;;; lP S;;; VK

and j rVK = id. It follows from Fact 1.5 that we may lift j to a new embedding

j : V[G] ~ M[G][H] S;;; V[G][H] .

Here we have denoted the new embedding by j also. There is no possibility ofconfusion because the new j extends the old one.

Notice that this embedding and its target are defined in V[G][H], a genericextension of V[G]. This is our first example of the notion of generic embeddingdefined in the last section. Notice also that N1 = NiIG] and K. = crit(j) = N~[Gl,while j(lI:) = N~[G][Hl.

SMALL CARDINALS 27

Let V[G]l= "8 is a stationary subset of TJ". It is easy to see that the canonical name for 8 is a member of VKH , and since VKH ~ M it follows that8 E M[G]. Since M[G] ~ V[G] and stationarity is downwards absolute, M[G] l="8 is stationary" .

We also have that j(8) E M[G][H], and since crit(j) = I\: it follows as in Fact1.4 that j(8) n I\: = 8. What is more, it follows from the countable closure of Q inM[G] that M[G][H] l= cf(K) =N}.

However we are still missing one thing; we need to know that 8 is stationary inM[G][H] before we can complete the reflection argument using the elementarity ofj : V[G] ----* M[G][H]. This problem is a very common one in arguments involvinggeneric elementary embeddings, for example we will find ourselves in an exactlysimilar situation in the discussion of the tree property in section 4 of this survey.

To finish the argument we use the following fact (really a special case of thefact that countably closed forcings are proper) .

Fact 2.3. Let 8 be a stationary subset of 8~, where I\: = cf(K) > w. Let lP' becountably closed. Then II-p "8 is stationary".

Proof. Let P E lP' be any condition and suppose that P II- "C is club". Let 0 besome very large regular cardinal and let <0 be a well ordering of Ho. Find a modelN ~ (Ho,E,<o) such that p,8,K,lP',C EN and 8 = Nnl\: E 8. Now choosean increasing sequence (8i : i < w) of elements of N n I\: which is cofinal in 8, anddefine (Pi : i < w) a decreasing sequence from lP' n N as follows; Po = P, and Pi+}is the <o-least condition such that Pi+} S Pi and Pi+} forces some ordinal largerthan s, into C.

Because Pi+} is defined from the parameters 8i,Pi, C, lP', we can see (inductively)that each Pi E N. If {Ji is the least ordinal greater than 8i which Pi+} forcesinto C then by a similar argument {Ji EN, so in fact {Ji E N n I\: = 8 and soPi+} II- Cn (8i, 8) f; 0.

Now use the countable closure of lP' to find Pw such that pw ~ Pi for all i < w.Clearly Pw II- 8 E lim(C) , so we have produced a refinement of P which forces amember of 8 into C. It follows that II-p "8 is stationary". 0

Using this fact we can conclude that M[G][H] l= "8 is stationary", and thenwe can argue exactly as in Fact 1.4 that by elementarity

V[G] l= "there is a E T'f such that 8 n a is stationary".

We promised at the start to show how the argument works from the weakerassumption that I\: is weakly compact. To do this, suppose that

II-p "8 is a non-reflecting stationary subset of TJ"

for some canonical name 8. Since 8 E H K+ we may find M a model of enough settheory such that IMI =1\:, <KM ~ M, 8 EM. We may also assume that.for everya E 8~w the model M contains a lP'-name for a club in a disjoint from 8 .

Now by the weak compactness of I\: there is k : M ----* N with crit(k) =1\:. If G is lP'-generic over V and H is k(lP')-generic over V then in V[G][H] we

28 JAMES CUMMINGS

get an embedding k : M[G] -+ N[G][H] by the same arguments as above. LetS = SG, then V[G] 1= "S is stationary". By Fact 2.3 we then get V[G][H] 1="s is stationary" and so a fortiori N[G][H] 1= "s is stationary" .

Now we can argue as before that M[G] 1= 3a E Tl Sn a is stationary. This isa contradiction, because we built names for clubs disjoint from sn a into M forevery a < K, with cf'(o) > w. 0

Notice that there is a problem with generalizing the proof of Fact 2.3 to largercofinalities, for example it is not obvious that N2-closed forcing will always preservethe stationarity of stationary subsets of S~l ' The problem is that when we buildthe chain of conditions p we may wander out of the structure N at limit stages.A crude solution to this problem is to make a cardinal arithmetic assumption andthen work with suitably closed substructures.

Fact 2.4. If K, = A+ and A<It = A then every J.L+ -closed forcing preserves thestationarity of stationary subsets of S; .

Sketch. Build N containing everything relevant such that INI = A, <It A = A andN n K, E K, . Then build a decreasing J.L-sequence from JPl n N as in the proof of Fact2.3; the closure of N makes the construction go through. 0

It follows that Baumgartner's theorem generalizes to any successor of a regularcardinal.

Theorem 2.5 (Baumgartner). Let A = cf(A) < K, where K, is weakly compact.If G is Oout); < K,)-generic then

V[G] 1= "If S ~ S~: is stationary, there is a E st with S n a stationary".

Sketch. In V[G] we have K, = A+ and A<>' = A. Thus we can mimic the proof ofTheorem 2.2, using Fact 2.4 in place of Fact 2.3. 0

Notice that we are immediately in difficulties if we try to generalize theseresults to successors of singular cardinals; one problem is that there is no obviousanalogue of the Levy collapse to make a large cardinal become the successor of asingular cardinal, another is that since N~o > Nw the trick of working with closedsubstructures will no longer work. As we see shortly, the problem is not merely atechnical one.

One subtle point is worth mentioning here. Inspection of the proof of Theorem 2.2 shows that actually the Levy collapse of a weak compact to N2 gives amodel in which any N1-sequence (Si : i < N1) of stationary subsets of TJ reflectsimultaneously (that is there is f3 E Tl such that S, n f3 is stationary for every i).

Jensen [20] proved that "every stationary subset of TJ reflects to a point ofTf" requires a Mahlo cardinal, Magidor [28] showed that "every pair of stationarysubsets of TJ reflect simultaneously to a point of Tl" needs a weakly compact cardinal, and Harrington and Shelah [18] showed that consistency of "every stationarysubset of TJ reflects to a point of Tf" follows from that of a Mahlo cardinal.

It is also possible to show that instances of stationary reflection for differentcofinalities are highly independent. In [5] models are constructed in which every

SMALL CARDINALS 29

stationary subset of TJ reflects and every stationary subset of Tr has a nonreflecting stationary subset, and vice versa.

The problem of stationary reflection has a much different flavor at successorsof singular cardinals. Here it is possible for every stationary set to reflect, but thishas a much higher consistency strength than that of a weak compact cardinal.

Fact 2.6. If K, < JL = cf(JL) and K, is u-supercompact then for every stationaryS ~ S~K. there is a E S~K. such that S n a is stationary.

Proof. Fix j : V -t M such that crit(j) = K" j(K,) > JL and IJM ~ M. Let'Y = sup(j"JL), then j"JL E M and thus M 1= cf(')') = JL. Since JL < j(K,) < j(JL) andj (JL) is regular in M, 'Y < j (JL). We claim that M 1= "i (S) n 'Y is stationary".

Let C ~ 'Y be club in 'Y. Let D = {a < JL I j(a) E C }, then it is routine tocheck that D is < x-club in JL. Since S ~ S~K.' this implies that D n S i- 0, and ifa E D n S then j (a) E C n i"S ~ C n (j (S) n 'Y).

Since M 1= "cf(')') < j(K,) and j(S) n'Y is stationary" it follows by elementaritythat there is a E S~K. such that S n a is stationary. 0

Actually, K, being JL-strongly compact would suffice here. As long as we areonly concerned with stationary sets of cofinality w ordinals the following result ofShelah [2) says that we can have reflection everywhere.

Fact 2.7. Let K, be supercompact. Let lP' = Coll(N ll < K,) . In VP, for every JL =cf(JL) ~ K, and every stationary subset of S~ there is a < JL such that cf(a) = N1

and S n JL is stationary.

The proof combines the ideas of Theorem 2.2 and Fact 2.6.

Fact 2.8. Let A be a singular limit of A+-supercompact cardinals, then every stationary subset of A+ reflects .

Proof. >.. is singular , so Va < >..+ cf(a) < >... If S ~ >..+ is stationary then theremust be a >..+-supercompact K, < >.. such that S nS~: is stationary, and then thereis a E S~: such that S n a is stationary. 0

It is natural to ask whether we need such strong hypotheses to get stationaryreflection at the successor of a singular cardinal . The exact strength needed is stillnot known, but we will see that it must be considerable.

Jensen [20] introduced the combinatorial principle 0>.: it states that thereexists (Co: : a < A+) such that Co: is club in A, o.t .(Co:) ::; A and (3 E lim(Co:)==> Cf3 = Co: n {3. The connection between 0>. and stationary reflection is thefollowing useful fact .

Fact 2.9. If 0>. holds and S ~ A+ is a stationary set, then there exists a stationaryT ~ A+ such that T n a is nonstationary for every a < A+.

Proof. Find T ~ Sand (3 such that T is stationary and Va E T o.t .(Co:) =(3. Nowif a < A+ and cf'(o) > w, then lim(Co:) is club in a and 'Y E lim(Co:) ==> C; =Co: n'Y. It follows that 'Y 1----+ o.t .(C-y) is I-Ion lim(Co:), hence Ilim(Co:) nTI $ 1and so T is nonstationary in a . 0

30 JAMES CUMMINGS

Jensen [6] proved that

- For all >., L F 0>..- If OU does not exist then for every singular >. we have >.t = >.t, from which it

follows that V F 0>. for every singular >..Combining these results, it follows that to have a singular cardinal such that everystationary subset of the successor reflects will require at least the strength of OU.

This argument has been greatly generalized by various workers in innermodel theory. Combining fairly recent results of Mitchell, Schimmerling, Steeland Woodin we get

Fact 2.10. If>. is singular and every stationary subset of >.+ reflects then Projective Determinacy holds. In particular for every n there is an inner model of "ZFC+ there exist n Woodin cardinals".

Another natural question is whether small successors of singulars such as NWHcan exhibit the phenomenon of stationary reflection. This question is answered bythe following result from [28] Magidor originally had a more complex constructionwhich involved more forcing, Shelah pointed out that the last step of Magidor'soriginal construction was not necessary.

Theorem 2.11 (Magidor [28]).Let (Kn : n < w) be an increasing w-sequence of supercompact cardinals. Definea forcing iteration: 1P'1 = Coll(w, < Ko) , IP'nH = IP'n * COll(Kn-l, < Kn)vlP n for1 ~ n < w, IP'W is the inverse limit of the IP'n' Then

V P", F "If S ~ NWH is stationary, there is a < NWH with S n a stationary".

We will sketch the proof and refer the reader who wants more details to [28].

Sketch. Let>' = sup; Kn; it is not hard to see that in V[Gw] the cardinal Knbecomes Nn+1 , >. becomes Nw and >.+ becomes NwH .

Let Gw be IP'w-generic over V . For each n there is a generic extension V[Gw][Hn ]

of V[Gw ] such that

- There is kn : V[Gw] ~ Mn ~ V[Gw][Hn] a generic embedding with criticalpoint K n .

- Hn is generic for Nn-closed forcing.- kn r>.+ EM, kn(Kn) > >.+.

In Foreman's terminology from [9], Kn is generically supercompact. kn is actually anextension of an embedding i« : V ~ M witnessing that K n is >.+-supercompact.

To complete the argument in the style of Theorem 2.2 and Fact 2.8 we need toargue that in V[Gw] the stationarity of a stationary subset of T~tl is preserved byNk-closed forcing. This is false in general by results of Shelah [38], but fortunatelyit is true in V[Gw ]. To see this we introduce Shelah 's notion of an approachableset.

Definition 2.12 (Shelah [38]). Let S be a subset of J.L where J.L = cf(J.L) > w.Then S is approachable iff there exists (xa : a < J.L) and a closed unbounded setC ~ J.L such that for every a E S n C there is c ~ a club in a such that o.t.{e) =cf(a) and "1[3 < a 31 < a en[3 = x'Y '

SMALL CARDINALS 31

Fact 2.13 (Shelah). Let"( = cf("() < {L = cf({L) . If 8 ~ 8~'Y is an approachablestationary set and IP' is "(-closed then 8 is still stationary in VP.

Proof. Let x and C witness that 8 is approachable. Let P be any condition inIP', and suppose P II- -o is club in {L" . Let N ~ (He, E, <e) be a structure whichcontains everything relevant, with the property that a =N n {L E en 8; fix c ~ asuch that V{3 < a 3"( < a c n {3 = x'Y '

The key point is that because x E N and a ~ N, we have c n {3 E N for all{3 < a . Now we build a descending chain of conditions (Pi : i < cf(a)) such thatPo =P and Pi is the <e-Ieast condition such that

- Pi ~ Pi for all i < j .- Pi forces some ordinal greater than the r: element of c into b .

If j < cf(a) then N can compute (Pi : i < j) from c n{3 for {3 the jth element of c,so (Pi : i < j) E N and thus the sequence pnever wanders out of N.

The proof now concludes exactly as the proof of Fact 2.3 does. 0

Shelah observed that NW+l is approachable in the model V[Gw]. Given this, wecan finish the proof of the result as follows.

Let V[Gw] 1= "8 ~ NW+1 is stationary" . Then 8 n T~;t-l is stationary for somen < w. Forcing with some Nn-closed forcing we get a generic embedding

such that crit(kn) = "'n, kn r-\+ E Mn and kn("'n) > -\+ .If we now let "( = sup kn ,,-\+ then "( < kn (-\+) and M 1= cf("() = Nn. 8 is

stationary in V[Gw][H] because NW+l is approachable in V[Gw] i it follows thatj(8) n"( is stationary in Mn and we can finish the argument exactly as in theproof of 2.3. 0

It is worth noticing that 0>. implies that A+ is approachable. For more on theconnections between squares and approachability see [13] and [4].

An important topic not touched on here is that of stationary reflection in [X]NO,where X is an uncountable set and [X]No is the set of its countable subsets . See[14] and [12] for more on this.

3. Saturated ideals

Suppose that", =cf(",) > w. By an ideal on", we always mean an ideal which isx-complete, normal and uniform.

Definition 3.1. Let I be an ideal on «. Then I is saturated iff the Boolean algebmP«]I has the ",+ -c.c .

Saturated ideals are closely connected with generic elementary embeddings;the basic results are due to Solovay [39] and Kunen [23].

We start by outlining Solovay's analysis of a saturated ideal.If I is any ideal then forcing with Pn]I adds U an ultrafilter on P", nV, with

the property that U n I = 0. The idea is to take an ultrapower of V by U, in

32 JAMES CUMMINGS

essentially the same way that Scott [35J took an ultrapower of V by a measure ona measurable cardinal.

If I,9 are two functions in V with domain K" then we define I -:::=u 9 <==:}

{ 0 I1(0) =g(o) } E U; this is an equivalence relation. Working in V[U] we define V" IU to be the set of equivalence classes and also define [J]Eu[g] <==:}

{ 0 I1(0) E g(o) } E U. The structure (V" IU, Eu) is called the generic ultrapower of V by U, and the standard proof of Los' theorem shows that for anyformula ¢J in the language of set theory

(V"IU,Eu) F ¢J([ft]U, •. . [Jn]U) <==:} {o I¢J(ft(o), . . •ln(o))} E U.

We get an elementary embedding j : (V, E) ---t (V" IU, Eu) by defining j(x) to bethe class of the constant function with value x.

For a general ideal I there is no guarantee that the structure (V" IU, Eu) iswell-founded. However if I is saturated then Solovay proved this will be the case,using the following key fact.

Fact 3.2 (Solovay). If Irp,,/I j E V,dom(j) = K, then there is 9 E V such that

Ir j -:::=u g .

Using this it is possible to show

Fact 3.3 '(Solovay). Let I be a saturated ideal on K, . Let U be an ultrafilter addedby forcing with PK,II . Then

- (V" IU, Eu) is well-founded, so can be identified with its Mostowski collapseto give a generic embedding j : V ---t M ~ V[U], where M -:::= V"IU.

- crit(j) = K, .

- V[U] F "M ~ M .In particular, if K, = Nl then we get an embedding such that j(N l ) = N2 =

Ntt = Ni[U1, where V[U] 1= wM ~ M . Notice that here V[U] 1= <i(~t> M ~ M (onemight say that Nl is generically almost huge, see Definition 3.4).

Kunen showed [23] that it is possible to go in the other direction, and deducethe existence of a saturated ideal from that of an appropriate generic embedding.In particular he gave the first consistency proof for the existence of a saturatedideal on Nl , starting from the consistency of a huge cardinal. Magidor [27J showedthat Kunen's argument can be made to work from an "almost huge" cardinal, andwe will outline this version.

Definition 3.4. K, is almost huge iff there exists j : V ---t M such that crit(j) = K,

and V F <i(,,) M ~ M.

, Let K, be almost huge and fix j : V ---t M such that crit(j) = K" j(K,) = A,<AM ~ M .

- We start by collapsing K, to Nl and A to N2 , to get a new model 1'1 in which2~o = Nl = K, and 2N1 = N2 = >..

- In Vl there is a 2-step forcing iteration IF *Q such that

• If G * H is IF * Q-generic then there is an extension of the original i ,

SMALL CARDINALS 33

• IP' is N2-c.c. and A = Nr1 = Nit .

• Q is N1-closed in vr.- Using the closure of Q in vr. it is possible to show that in vr we can define

an ultrafilter on P« n V1 • The key points are that V1 1= 2N1 = N2, vr 1=IPK, n Vii =N1, and vP 1= "Q is countably closed" . Using this we can work invt and build a decreasing chain of conditions to decide "K, E j(X)" for eachX E PK, n V1 • Let if name this ultrafilter.

- Working in V1 , let I = { X ~ K, I Hoop X ~ if }. Using the N2-c.c. of IP' in Vt, itis possible to show that

V1 1= "I is a saturated ideal on N1".

This style of argument will serve to get saturated ideals on many cardinals.The culmination of this line of development is Foreman's paper [8] in which it isproved to be consistent that every regular cardinal should carry a saturated ideal.Foreman and Laver [11] showed that is also possible to get stronger forms of chaincondition for the quotient algebra.

However some questions were left open: for example

- How strong is the existence of a saturated ideal on N1?- Can the non-stationary ideal on N1 be saturated?

For a time it was conjectured that an almost-huge cardinal was the right assumption to get a saturated ideal. Foreman, Magidor and Shelah's work [14] onthe forcing axiom MM (Martin's Maximum) showed that this is not the case; in[14] it is shown (among other things) that

- Con(ZFC + there exists a supercompact cardinal) implies Con(ZFC + MM).- MM implies that the nonstationary ideal on N1 is saturated.

The existence of an almost huge cardinal is known to be a much stronger assumption than the existence of a supercompact.

The question of the strength of a saturated ideal on N1 is now almost settled,in the light of the following results.

Fact 3.5 (Steel [41]). If the nonstationary ideal on N1 is saturated and thereexists a measurable cardinal then there is an inner model of "ZFC + there existsa Woodin cardinal".

The assumption of the existence of a measurable cardinal is a technical devicehere. It is conjectured that the saturation of the nonstationary ideal should suffice.

Fact 3.6 (Shelah [37]). If 8 is Woodin then there is a forcing extension in whichN1 is preserved, 8 becomes N1, and the nonstationary ideal on N1 is saturated.

We outline the proof of Shelah's result. We require Shelah's concepts of semiproperness and revised countable support iteration, for which we refer the readerto Goldstern's paper [17] in this volume.

Definition 3.7. Let A be a maximal antichain in PNdNS, where NS is thenonstationary ideal on N1 • Then we define a poset §(A) as follows: (f, c) E §(A)iff

34 JAMES CUMMINGS

dom(J) =max(c) < N1 .

rge(J) ~ A.c is closed.

- 'tIf3 E C 30: < f3 f3 E 1(0:) ·

The ordering is extension.

§(A) is defined in [14], and is used there to show that MM implies the saturationof NS. For any A it will be the case that §(A) is stationary preserving. §(A)makes IAI = N1 and shoots a club through the diagonal union of A, from whichit follows that A will be a maximal antichain of size N1 in any extension of VS(A)

by stationary preserving forcing.

Definition 3.8. A is a semiproper antichain iff §(A) is semiproper.

We can now outline Shelah's argument; essentially the idea is to force exactlythat fragment of MM which is needed to get the saturation of NS. We start byassuming that 0 is a Woodin cardinal.

- The construction is a revised countable support iteration of length 0, whereat stage 0: we force with § (Ao:) *Coil (N1, 22Q

) for some Ao: such that vPQ 1="Ao: is semiproper" . The Ao: are chosen using some kind of diamond principle.

- At the end of the construction we have a semiproper forcing 1P'c5, which willpreserve N1 and make 0 into N2 . V P

6 1= 2~1 = N2 • We need to check that thenonstationary ideal is saturated.

- Let V P6 1= "(Ao: : 0: < 0) is an antichain in PNI/NS". Applying the Woodin

ness of 0 and the diamond principle used in defining the iteration we findK, < 0 such that

VP" 1= "(Ao: : 0: < K,) is a semiproper antichain"

and All: = (Ao: : 0: < K,) . At stage K, the antichain (Ao: : 0: < K,) is made maximal; it follows that every antichain in PNI/NS has size at worst Nl.

At the heart of the argument lies the idea of a structure "catching an antichain"which comes from Foreman, Magidor and Shelah's work in [14] . Let A E N -< Ho,where A is a maximal antichain in PNI/N S and N is countable. We say thatM 2 N "catches A" iff

- M n N1 =N n N1 ( =0 say).- There is A E A n M such that 0 E A.

Assume that M catches A, and that A E A is such that 0 E A . Suppose thatwe have some condition (P,c) E N n §(A) ; then if € = dom(p) we have e < O.Working in the standard way we can build a decreasing w-chain of conditions in§(A) nM which meets every maximal antichain of §(A) lying in M; this sequencewill have a lower bound because

- oEA.- A E M so that A gets enumerated before 0 by the first entry of some condition

in the chain.

The lower bound will be a weakly (N, § (A) )-generic condition because N n WI =MnWl.

SMALL CARDINALS 35

It is worth remarking that the combinatorics of this argument resurfaces inWoodin's theory [43] of the stationary tower forcing.

We have only scratched the surface of the subject of saturated ideals here. Weconclude by listing some of the other important results in the field.

Fact 3:9 (Woodin [44]). If the non-stationary ideal on Nt is saturated and thereis a measurable cardinal then 8~ = N2 (this is a strong form of the negation of theContinuum Hypothesis).

Fact 3.10 (Shelah [36]). If I is a saturated ideal on.x+ then (as a corollary ofa general result on changes of cojinality) { a < .x+ I cf'(o) # cf(.x) } E I.

Fact 3.11 (Gitik and Shelah [16]). If K, is weakly inaccessible then the nonstationary ideal on K, is not saturated. If K, is singular then the non-stationaryideal on K,+ restricted to { a < K,+ I cf(a) = cf(K,) } is not saturated.

Fact 3.12 (Woodin [44]). The following are equiconsistent

1. AD2. There exists an Nt -dense ideal.3. The non-stationary ideal on Nt is Nt-dense.

Fact 3.13 (Foreman [10]). From large enough cardinals it is consistent thatthere exist a Nt-dense countably closed weakly normal ideal on N2 •

4. The tree property

We recall a few basic definitions about trees (see [26] for more details).Let T be a tree, let K, be a regular cardinal.

1. T is a «-tree iff ITI = ht(T) = K, and 'Va < K, ITal < K" where Ta is the a t h

level of T.2. T is a K,-Aronszajn tree iff T is a x-tree with no cofinal branch.3. T is a special >..+ -Aronszajn tree iff there exists h : T -t >.. such that x <T

y ==> h(x) # h(y).4. K, has the tree property iff there is no x-Aronszajn tree .

The following easy argument gives a connection between elementary embeddings and the tree property. We write T t {3 for Ua<~ T~.

Fact 4.1. If K, is measurable then K, has the tree property.

Proof. Let T be a x-tree, and fix j : V --t M with crit(j) = K,. Then j(T) is aj(K,)-tree in M, and what is more j(T) r K, is isomorphic to T (here we use the factthat each level Ta has size less than K" so that j(Ta ) = j"Ta ) .

j (K,) > K, so j (T) has at least one point on level K,. Looking at the pointsbelow this point we see that j (T) r K, has a cofinal branch in M j since j (T) rK, isisomorphic to K, and M ~ V, T has a cofinal branch. 0

In fact K, being weakly compact would suffice here: just build T into an appropriate structure of size K,. It is known that the weakly compact cardinals areexactly those inaccessible cardinals which have the tree property. It is also known

36 JAMES CUMMINGS

that if >.<'\ = >. then there is a special >'+-Aronszajn tree, in particular CH givesa special N2-Aronszajn tree .

It is natural to ask whether there can be a model with no N2-Aronszajn tree. Anatural first try might be to take a measurable cardinal K and Levy collapse it toN2 ; this fails (because CH holds after doing the Levy collapse) but it is instructiveto see exactly what goes wrong.

Let K be measurable, let j : V ---t M be an elementary embedding withcrit(j) = K. Let G be Coll(wl, < K)-generic over V. Then as in Section 2 we geta generic embedding j : V[G] ---t M[G][H] ~ V[G][H], where H is generic overV[G] for the countably closed forcing Coll(Nt,j(K)-K). Now let us try and imitatethe proof of Fact 4.1. If V[G] 1= "T is an N2-tree" we can argue as before that Thas a cofinal branch in M[G][H] and hence in V[G][H]. We would like to arguethat T must have a cofinal branch in V[G], but at this point the argument failsbecause it is quite possible in general for countably closed forcing to add a branchto an N2-tree. For example if V = L there is a countably closed N2-Souslin tree 8,and then (8, ~s) is a countably closed poset which adds a branch through 8.

Mitchell resolved the problem by proving

Fact 4.2 (Mitchell [31]). The following are equiconsisieni.

1. There exists a weakly compact cardinal.2. N2 has the tree property.

We will sketch Mitchell's argument, but we begin by stating a couple of usefulfacts about trees and forcing.

Fact 4.3 (Silver) . If2No > Nt, countably closed forcing cannot add a new branchto an N2-tree.

Fact 4.4 (Kunen and Tall [24]). Let lP have the property that for every Nt

sequence of conditions from lP there is a subsequence of length Nt of pairwise compatible conditions (P is Nt-Knaster). Let T be a tree of height Nt with no cofinalbranch (not necessarily an Nt-tree) . Then forcing with lP cannot add a cofinalbranch through T .

We now give an outline of Mitchell's argument (this way of presenting theargument appears in Abraham's paper [1]). Once again we will assume that K isa measurable cardinal and indicate at the end how to weaken the assumption toweak compactness.

- Let j : V ---t M with crit(j) =K.- Define lP (we will not give the definition) a forcing with the following proper-

ties:

1. [P] =K, lP is K-C.C. and lP ~ VI( '

2. If f3 < K is inaccessible then lP,8 =def lP n Vp is a complete subforcing ofP, and in V P

j3 the quotient lP/lPp is a projection of Add(w, K- f3) X Qpfor some countably closed Qp. Moreover VP

j3 1= Nt = NY, 2No = N2 = f3and VPj3* Q j3 1= K=N2 •

3. vP 1= Nt = Ny,2No = N2 = K.

SMALL CARDINALS 37

- If G is IP'-generic then it is possible to lift j : V ---+ M to get a genericembedding j : V[G] ---+ M[G][H] . This is easy because IP' ~ V~, j r IP' = idp,and IP' is x-c.c.

- Suppose that V[G] l= "T is an N2-Aronszajn tree" . T E M[G] because T hasa name in V~+l and V~+l ~ M, and arguing as before T has a cofinal branchin M[G][H].

- We work in M[G] to do a factor analysis of j{lP')jG. By elementarity we seethat M[G] ~ M[G][H] ~ M[G][hd[h2], where hI is Q-generic over M[G] forsome Q which is countably closed in M[G], and h2 is Add(w,j(~))-generic

over M[G][hd .- Q adds no branch to T because M[G] l= 2No = N2 and Q is countably closed.

Q collapses ~ to be an ordinal of cardinality and cofinality NI, so if we takean NI -sequence cofinal in ~ and look at the corresponding levels of T we geta "squashed" tree T* which has height Nl and no cofinal branch.

- Forcing with h2 cannot add a cofinal branch to T* because Add(w,j(~))

has the N1-Knaster property. This is a contradiction because T has a cofinal branch in M[G][H] and M[G][H] ~ M[G][hd[h2].

- If ~ is only weak compact the argument is similar. Suppose T is a canonicalIP'-name and 11- "T is a x-Aronszajn tree". Build T into an appropriate M,and let k : M ---+ N with critical point K. As before we lift k to a new mapk : M[G] ---+ N[G][H]. N[G] ~ V[G] so N[G] l= "T has no cofinal branch" .By the usual elementary embedding argument T has a branch in N[G][H].This leads to a contradiction as before.

We conclude with a few remarks about other results on the tree property.

Fact 4.5 (Abraham [1]). If K is supereompact and A > ~ is weakly compact,there is a forcing extension in which 2No = ~ = N2, 2N1 = A = N3, and both N2and N3 have the tree property.

Fact 4.6 (Foreman and Magidor). If two successive cardinals have the treeproperty, there is an inner model with a strong cardinal.

Fact 4.7 (Magidor and Shelah [29]). From a very strong large cardinal hypothesis, it is consistent that NW+1 should have the tree property.

Fact 4.8 (Cummings and Foreman [3]). If it is consistent that there are wsupercompact cardinals, it is consistent that Nn has the tree property for every nwith 2 $ n < w.

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36. S. Shelah, Proper forcing, Lecture Notes in Mathematics 940 , Springer-Verlag,Berlin, 1982.

37. S. Shelah, Proper and improper forcing, to appear.38. S. Shelah, On successors of singulars, in: Logic Colloquium 78, (M. Boffa, D. van

Dalen and K. McAloon, eds.}, Studies in Logic and the Foundations of Mathematics97, North-Holland, Amsterdam, (1979), 357-380.

39. R. Solovay, Real-valued measurable cardinals, in: Axiomatic set theory, Proc. Sympos. Pure Math . 13, Part I American Mathematical Society (1971), 397-428.

40. R. Solovay, W. Reinhardt, and A. Kanamori, Strong axioms of infinity and elementary embeddings, Annals of Mathematical Logic 13 (1978), 73-116.

41. J. Steel, The core model iterability problem, to appear.42. S. Ulam, Zur Masstheorie in der allgemeine Mengenlehre, Fundamenta Mathemati

cae 16 (1930), 140-150.43. W. H. Woodin, Supercompact cardinals, sets of reals and weakly homogeneous trees,

Proceedings of the National Academy of Sciences of the USA 85 (1988), 6587-6591.44. W. H. Woodin, Forcing axioms, determinacy and the nonstationary ideal, to appear.45. D. Wylie, Condensation and square in a higher core model, Ph.D. thesis, Mas

sachusetts Institute of Technology (1990) .

COUNTABLE LENGTH RAMSEY GAMES

CARL DARBYDepartment of Computer Science, Mathematics and StatisticsMesa State CollegeGrand Junction, COUSA

AND

RICHARD LAVERlDepartment of MathematicsUniversity of ColoradoBoulder, CO 80309USA

Abstract. It is known that the first player has a winning strategy for the Ramseygame R(a, < n, a) if a < w2 is a limit ordinal, and for the Ramsey game R(a, 2, a)if a < Wl is a limit ordinal or the successor of a limit ordinal . By way of contrast, weshow that the second player wins R(wW,3,wW

) . More generally, the second playerwins R(cp,3,ww V (WW )*) for cp a scattered linear order type of any cardinality(where the game lasts w-many moves).

Let cp,'ljJ be order types, n < w. The Ramsey game R(cp,n,'ljJ) is played by twoplayers, White and Black, as follows. Let L be a linear ordering, tpL = tP. ThenWhite and Black, moving alternately with White moving first, play to choose asequence

with each Wi, b; a member of [L1n which has not been previously chosen. Whitewins if and only if there's an L' ~ L, tpL' ='ljJ, [L']n ~ {wm : m < w}. For x ::;wthe game R(ip, < n, 'ljJ) is played similarly: the players pick previously unchosenmembers of [L1<K (where tp L = cp) and to win White must have chosen all themembers of some [L'l<1I: with tp L' ='ljJ. Then

Theorem 1 (Baumgartner, Galvin, Laver, McKenzie [1]).

(i) Black wins R(w,< w,w).(ii) White wins R(w, < n,w) for all n.

IThis work was partially supported by NSF Grant DMS 9626713.

41

C.A. Di Prisco et al. (ed.), Set Theory, 41-46.© 1998 KluwerAcademic Publishers.

42 DARBY AND LAVER

(iii) White wins R(a, 2, a) if a < WI is a limit ordinal or the successor of a limitordinal.

In [IJ, the question was asked whether White wins R(a, < n, a) for all limita < WI, n < w. And the question of the status of longer length games (Whitemoving first at limit stages) on uncountable linear orderings, was raised. Thispaper is about the first of these questions.

McKenzie [6J proved that White wins R(w . 2,3, W • 2). The strongest suchpositive result at present is

Theorem 2 (Darby [2]). For all n, k < w, White wins the game R(w· k, < n, cak) .

It's not known whether for all n < W and A with w2 $ A limit < wW, Whitewins R(A, < n, A). We'll show that White does not win for A~ W

W• Recall that an

order type ip is scattered iff sp contains no subset isomorphic to the rationals.We will prove

Theorem 3. Let ip be scattered. Then Black wins R(cp,3,ww V (WW )*).

The theorem asserts that Black has a strategy guaranteeing that for no L'with tp £I =WW or tp L' = (WW

) * will [L'P ~ {wn : n < w}. If sp is countable thetheorem may be derived using Galvin's negative partition theorem for WWj for cpof arbitrary size (but remember the game still only lasts w-many moves) it will bederived from a game theoretic version of Galvin's theorem.

Regarding the second question, Hajnal and Nagy proved some results on longerlength games. Let R(cp, n, 'ljJ) be like R(cp, n, 'ljJ) except the players continue choosingmembers of an [LJn (tpL = cp) until all members of [LJn have been picked, withWhite moving first at limit stages.

Theorem 4 (Hajnal and Nagy [4]).

(i) If 2'No = N1 and Chang's Conjecture (N2 -t (Nd~~l'No) hold, then Black wins

R(N2 , 3, Nd.(ii) If K. is a singular strong limit, then White wins R(K., 2, K.) .

(iii) If K. -t ({3)~'No, 13 a limit ordinal, then White wins R(K., < w,(3) .

The abovementioned theorem that White wins the R(w · k, < n,w' k)'s uses amodification of the methods used to prove (iii) . The games R(cp,n, 'ljJ) considered

in [4J have the same outcome if the players only make ~-many moves. For moreresults and problems on uncountable length Ramsey games see [4] .

For L a linear ordering, let <Lex be the lexicographic ordering, and < theordering of proper initial segment, on (L) $w .

Call Y ~ (L)<w good if and only if for every f E (L)W there's an n < w suchthat f rn f. y for all y E Y . Recall (Hausdorff [5]) that the class of scattered typesis the closure of {a, I} under well ordered and converse well ordered sums . Then(by induction) a type cp is scattered and countable iff there's a good U ~ (Z)<wwith tp (U, <Lex) = tp, For convenience, we may, and will below, arrange that eachsequence in U is increasing in absolute value. Let

COUNTABLE LENGTH RAMSEY GAMES 43

Definition 5 (Galvin). If u, v E X the oscillation osc(u, v) is defined as follows.Let

e = {Inl :n E rangeu} .6. {Inl: n E rangev}.

Then assuming e i 0, e, written in increasing order, partitions into blocks: e =Bo U B1u ··· UBi, where the Bm's alternately are nonempty subsets of {Inl : n Erangeu} and {Inl :n E rangev} . Then osc(u,v) =i. (Let osc(u ,v) = 0 if e =0.)Theorem 6 (Galvin [3]). Let cp be a linear order type.

(i) IfG ~ X is good, tp (G, <Lex) = wn or (wn )*, then for each j with 1 $ j < 2n,there is a {u,v} E [GJ2 with osc({u,v}) = j .

(ii) If ip is scattered and countable, then sp It [WW V (WW)*J~o '

(iii) If ip is scattered and does not embed Wl , then

.s: [ 2 2 3 3 J2cp TT W,W ,W ,W ,W , • . • •

Remarks. (ii) follows from (i). The negative partition relation in (iii) asserts thatfor tp L = cp there's a coloring c: [Lj2 -t w, such that any subset of L of type W

contains a pair having color 1, any subset of L of type w2 contains pairs havingcolors 2 and 3, etc. Part (iii) follows from a version of part (i) where G is a goodsubset of (Ord" Uw)<w and the oscillation between two members of G is taken tobe the oscillation of their positive coordinates. More general results on oscillationsbetween arbitrary sets of integers, and a proof of Theorem 6, appear in Todorcevic[7J .

Let H be the following game. White and Black playa sequence wo, bo, Wl, b1,

. . . ,Wn , bn , ... for n < w, where each Wi E (11, each bi E X, and Wi < Wj {:}b, <Lex bj. White (lest he lose) must choose so that {Wi : i < w} is well ordered.Assuming {Wi : i < w} is well ordered, Black wins if and only if {bi : i < w} isgood.

Lemma 7. Black wins H (by a strategy ensuring that b, 1:- bj (all i i j)).

Proof. For s, t E X let s /\ t be the greatest common initial segment of sand t.Black plays so that Wi < Wj {:} b, <Lex bj, and all of the following hold :

(i) bi 1:- bj all i,j .(ii) For sEX, min, at most one of s~m, s~n belongs to {bi : i < w}.

(iii) If j < w and Wj < Wi (all i < j) (respectively, Wj > Wi (all i < j)) , thenbj(O) < bi(O) (all i < j) (respectively, bj(O) > bi(O) (all i < j)) .

(iv) Suppose i , k < i. Wi < Wj < Wk and for no j' < j is Wi < Wjl < Wk. Theneither (bj /\ bi) or (bj /\ bk) has length 1 + length (bi /\ bk).

Black does this as follows. Let bo be any member of X of length > 1. Assumewo,bo, ... ,Wj-l,bj-1,wj have been picked. Ifwj < Wi (all i < j) or Wj > Wi (alli < j), pick bj of length> 1 to satisfy (iii). Suppose Wi < Wj < Wk are as in thehypothesis of (iv). By (i) and (ii), at least one of bi,bk has length ~ 2 + length (bi/\bk). Suppose it's b, (the case of bk being similar). Then bi = (bi 1\bk)~mO'""'ml '""' s,for mo,ml E Z, with s perhaps empty. Let bj = (bi /\ bk)'""'mO'""'m2'""'m2 +1, wherem2 = Imll + 1. Then (i)-(iv) and the order isomorphism of the map W n -t bn stillhold .

44 DARBY AND LAVER

Suppose for a contradiction that Black does not win this play of the game;then {Wi : i < w} ,and thus {bi : i < w}, are well ordered but {bi : i < w} is notgood. Pick I E (Z)W such that for each n there's an i with I rn < bi. By this and(i), no bi < I.

Case 1. For all i, i. <Lex ILet j be least such that bi r {OJ = I r {OJ. Then for all i < i, bi <Lex bi.

Let £ > j be least with bi >Lex bi . By (iv), bi(O) > bi(O) = 1(0), contradictingbi <Lex I·

Case 2. For some i, b, >Lex I .There must be a largest initial segment of I, call it c, such that c < b, for some

b; >Lex f. Namely if no largest such c existed, there'd be an infinite descendingsequence in ({bi : i < w}, <Lex). Nowfix i such that bi is <Lex-least with bi >Lex Iand I A b, = c. Pick d with c < d i be least such that d < bi. Thenbi <Lex bi, and for no n :::; £ is bi <Lex b« <Lex bi. Let £' > £ be least suchthat bi <Lex bi' <Lex bi. Then by the definition of c and the minimality of bi,bi' A I ~ bi AI. This contradicts (iv) for the triple bi, bi', bi. This completes theproof of the lemma. 0

Let H* be the same as H except White must make {wn : n < w} converselywell ordered. Then note that Black's winning strategy for H is also a winningstrategy for H* (since bi I. bi for all i and i, the argument for H* is symmetrical) .

In contrast to Lemma 7, White wins the version of the game H where he isonly required to make {wn : n < w} scattered-it is seen that he may play Wo = 0,WI = 1, W2 = 1/2 and then continue picking left or right midpoints to determine ashrinking sequence of intervals (thus {wn : n < w} embeds into w + w*) in such away that {bn : n < w} is not good. However, for the Ramsey game application we'llonly need a win for Black when White picks from a fixed scattered ordering. For I{)

an order type let HIP be the following game. Let tp L = I{). White and Black playwO,bO,WI,bl, . . .wn,bn, . . . (n < w), where Wi E L, b, E X, Wi < Wi ¢> b, <Lex bi.Then Black wins if and only if {bi : i < w} is good.

Theorem 8. For I{) scattered, Black wins HIP.

Proal. If I{) is a well ordering (or converse well ordering), then Black wins HIP bya strategy which ensures that bi I. bi for all i,j . Namely, Black embeds White'smoves into Q as the game proceeds and plays according to the winning strategyfor H (or H*) of Lemma 7.

For I{) an arbitrary scattered type, proceed by induction on Hausdorff's hierarchy. Assume cp = L: epa, where Black wins each H",o. Let tpL = I{), inducing

a<,.L = U La, tpLa = epa ·

a<,.Let aa be a winning strategy for Black for H",o' playing on La. Let a be a

winning strategy for Black for H,. ensuring that b, I. bi for all i =P j.

COUNTABLE LENGTH RAMSEY GAMES 45

A sequence of White moves (wn : n < w), wn E L, in H", induces a (possiblyfinite) sequence (ao, aI, .. . ) of White moves in H", : Win ELan' where io =0 andin+! is the least m > in such that W m ¢ Lao U . . . ULan'

Black's strategy for H", is as follows. Suppose Wo, bo, WI, bl , . . . , Wn havebeen played, with Wn E La. Let (ao, aI, . 00 , aj) be the moves in H", induced by(WO,Wl,oo "Wn ) ; so a = ai for some i ~ j. Let (wg,wf,oo.,wf) be the subsequence of (wo,WI,. '" wn ) consisting of those we's belonging to La; so wf = Wn .

Let Ca = CT( (ao, al, . .. , ai}). Let d = CTa ( (wg, wf , 00 • , wf}) . Shift d away from Caas follows. Let m = sup{I£1 : £ E range (ca )} + 1. Then for d = (uo,U1, . . . ,up) letd = (u~, ui, . .. , u~) where u~ = Ut + m if Ut ~ 0, u~ = U t - m if Ut < O. Thenlet Black's move be bn = Ca ~d. Using that a f: 13 => Ca f. c{3, it is seen thatW n < W m ¢:> bn <Lex bm, and ibn : n < w} is good.

This completes the proof when ip = L: CPa; the case ip = (L: CPa) * is similar.a<", a<",

oProof of Theorem 3. Let tp L = ip, As the players play Wo, bo,WI ,b1, . . . (Wi, b, E

[L]3), Black, using his winning strategy CT for H"" assigns to each £ E U(wn U bn )n

an i c x, so that £ < £' ¢:> i <Lex i' and {i: £ E U(wn U bn )} is good.n

Fix well orderings <x of X, <[xj2 of [XF, each of type W. Suppose Wo, bo, WI,b1, . . . , Wn have been played. We describe Black's next move. If Wn - (wo U boU... U Wn-l U bb-d f: 0, then Black, using CT, first assigns an i to each £ in thatset. Write W n = {£0'£1'£2} with io <x il <x i2' Let osc(il,l2) =p. Let {Yo,Ydbe the pth member of [XF in the ordering <[Xj2. Suppose that Yo = leo, Yl = k1where ko,k1 E Wo U bo U 00. U Wn . Then if one of {ko'£1'£2}' {k1,£1,£2} is not amember of {wo, bo, .. . ,Wn }, Black picks bn to be that triple (choosing arbitrarilyif both choices are possible). If the above suppositions don't hold, Black picks b«to be an arbitrary previously unchosen member of [L]3.

To see that Black wins, assume for a contradiction that there's a K ~ L, K oforder type WW or (WW )* , with [K]3 ~ {wn : n < w}. Let K = {k: k E K}. Pick{leo, kd E [KF. Then for some p, {ko,kd is the pth member of [X]2 under <[X)2.

For £ E L let ne be the least n such that £ E W n U bn . Let K' = {k E K : ko,k1 <xkand nko, nkl < nk}. Then K - K' is finite so K' has order type WW or (WW

) *. AndK' is good since K' ~ {l :£ E U(Wi Ubi)} and that set is good.

i

Thus there are il,l2 E K' with osc({il,l2}) = p. We are assuming that{ko,£1,£2}, {k1' £1,£2} E {wn : n < w}. But at the first place in the game whereWhite picked a {k ,£1,£2} with k <x II <x 12, Black responded by picking oneof {ko, £1, £2} or {k1,£1,£2} (if he had not previously picked both those triples).Contradiction. 0

References

1. J. Baumgartner, F. Galvin, R. Laver and R. McKenzie, Game theoretic versions

46 DARBY AND LAVER

of partition relations, Colloquia Math. Soc. Janos Bolyai 10, Keszthely, Hungary,131-135.

2. C. Darby, Ph.D. Thesis, University of Colorado (1990).3. F. Galvin, Circulated lecture notes (1971).4. A. Hajnal and Zs. Nagy, Ramsey Games, Transactions of the American Mathemat

ical Society 284 (2) (1984) 815-827.5. F. Hausdorff, Grundziige einer Theorie der qeordnetet: Mengen, Math. Ann. 65

(1908) 435-505 .6. R. McKenzie, Private correspondence, (1974).7. S. Todorcevic, Oscillations of sets of integers, preprint.

WEAK FORMS OF THE AXIOM OF CHOICE AND PARTITIONSOF INFINITE SETS

OMAR DE LA CRUZDepartment of MathematicsUniversity of FloridaGainesville, FL 32611USA

AND

CARLOS AUGUSTO DI PRISCOInstituto Venezolano de Investigaciones CientificasApartado 21827Caracas 1020-A , Venezuela

Abstract. There are many equivalent versions of the Axiom of Choice, and alsomany interesting consequences which are strictly weaker in terms of consistencystrength. We examine various of these weaker forms of the Axiom of Choice andstudy how they are related to each other.

1. Introduction

The Axiom of Choice and its role in mathematics is one of the most interestingaspects of the development of Set Theory. This axiom asserts the possibility ofpicking an element from each of the sets in an infinite collection without establishing a specific rule determining the choices. The Axiom was stated by Zermeloas follows:

AXIOM OF CHOICE (AC): For every family F of non-empty sets there is a function f : F --+ UF such that for every X E F , F(X) EX.

It is well known that AC is not provable from the rest of the axioms of settheory, and that it is consistent with them provided they are not contradictory.

Weak forms of the Axiom of Choice have attracted the interest of mathematicians for several reasons. Even if the use of the full Axiom is widely accepted,sometimes it is convenient to know how much of its strength is really needed toobtain a desired mathematical result. Moore [18] has given a very complete historical survey about the Axiom of Choice and its different versions. H. and J.

47

C.A. Di Prisco et al. (ed.), Set Theory. 47-70.© 1998 Kluwer Academic Publishers.

48 DE LA CRUZ AND DI PRlSCO

E. Rubin, in [23] and [24] , present many equivalent versions of the Axiom. Thefollowing choice principles are consequences of AC:

SELECTION PRINCIPLE (SP): For every set X there is a function assigning aproper non-empty subset to every subset of X with at least two elements.

THE ORDERING PRINCIPLE (OP): For every set X there is a linear order for X.

AXIOM OF CHOICE FOR WELL ORDERABLE SETS (ACWO): For every set Xthere is a function assigning one of its elements to every non-empty wellorderable subset of X.

AXIOM OF CHOICE FOR FINITE SETS (ACF): For every set X there is a function assigning one of its elements to each finite non-empty subset of X.

AXIOM OF CHOICE FOR FINITE SETS OF n ELEMENTS (Cn ) : For every set Xthere is a function choosing one element from each n-element subset ofX.

THERE ARE NO AMORPHOUS SETS (PP): Every infinite set X can be partitionedinto two infinite subsets.

Theorem 1.1. The following implications are provable in ZF, and none of thearrows can be reversed.

AC => SP => OP => ACF => VnCn => Cm => PPAC => ACWO => ACF

Proofs for these strict implications can be found in [11] or [18]. To establishthe strict character of the implications permutation models and symmetric modelsare used . In recent years several interesting results obtained using these techniqueshave appeared (See [6], [8], [9], [21], [28]) .

In the next section we explain briefly the techniques involved in the construction of permutation models and symmetric models . In section 3, we develop somerestricted and diminished versions of the choice principles listed above. In Section 4, we describe the cylinder model (a permutation model) and its properties,particularly with regard to these choice principles. In Section 5, we treat the independence of and implications between various choice principles. In Section 6,we add Ramsey's Theorem to our discussion of choice principles. While symmetricmodels satisfy ZF, permutation models satisfy ZFA; in the final section, we discusstransfer of some of the independence results for ZFA obtained using permutationmodels to corresponding independence results in ZF.

2. Permutation and Symmetric Models

The first independence results related to the Axiom of Choice were obtained byA. Fraenkel in 1921 using permutation models . These are models of the theoryZFA, Set Theory with Atoms. In many cases independence results in ZFA canbe turned into independence results in ZF by using some techniques developed byJech and Sochor [11] and also by Pincus [20] (see section 7).

WEAK FORMS OF AC 49

P. Cohen created the method of forcing and the idea of generic extensions ofmodels of ZFC to prove the independence of the Continuum Hypothesis from theaxioms of ZFC. To prove the independence of AC from the axioms of ZF, heintroduced the notion of symmetric submodel of a generic extension.

We will present here just a quick introduction to permutation and symmetricmodels. For a more extensive development of the theory ZFA, and of permutationand symmetric models we refer the reader to [l1J or [12J . Truss' paper [27J surveysconnections between permutation groups and the axiom of choice. In particular itexamines the finite versions of choice mentioned above and the non-existence ofamorphous sets .

The language of ZFA is the usual language of ZF with two constant symbolsA and 0 which denote respectively the set of atoms and the empty set . The axiomsof ZFA are the axioms of ZF with the following modifications:

- Empty Set Axiom:-.3x(x E 0)

- Axiom of Atoms:Vz[z E A +-t z ¥= 0 t\ -.3x(x E z)]

and the following modified versions of axioms of ZF:

- Extensionality:

('Ix ¢ A)(Vy ¢ A)[Vu(u E x +-t u E y) +-t x = yJ

- Regularity:Vx[(x ¥= 0 t\ x ¢ A) ~ 3y E x(x n y = 0)J

For a given set A of atoms, define

pO(A) =POl+l(A) =

PA(A) =

A

POl(A) u P(POl(A))

UpOl(X), for X limitOlEA

U POl(A) .OlEOn

If A is transitive, POO(A) and each POl(A) are transitive. The Axiom of Regularity implies that every set is in V.

The subclass POO(O) c V, formed by all those sets with no atoms in theirtransitive closure, is a model of ZF and is called the kernel.

Given a group 9 of permutations of A, for each x let

symg(x) ={1l' E g :1l'X =x}

Clearly, symg(x) is a subgroup of g.

50 DE LA CRUZ AND DI PRISCO

Definition. Given any group g, a set F of subgroups of 9 is a normal filter in 9iffor any H, K subgroups of 9 it holds:

i) 9 E Fii) If HE F and H c K , then KEF

iii) If H E F and KEF, then H n KEFiv) If 1T E 9 and H E F , then 1TH1T-1 E F

2.1. PERMUTATION MODELS

To define a permutation model we start with a base model V, with a set of atomsA, satisfying ZFA + AC + A is infinite. Given a group 9 of permutations of Aand a normal filter Fin Q, x E V is said to be symmetric if sym(x) E :F. Thepermutation model is the class

v = {x : x is symmetric and x c V}

of hereditarily symmetric sets.Permutation models defined using the normal filter generated by an ideal of

subsets of A are of particular importance.

Definition. A family I of subsets of A is a normal ideal if for all subsets E andF of A,

i) 0 E I,ii) if E E I and F ~ E then F E I ,

iii) if E E I and F E I then E u F E I,iv) if 1T E 9 and E E I , then 1T" E E I,v) for each a E A, {a} E I.

For every x, let

fixg( x) = {1T E 9 : 1TY =Y for all y EX};

fixg(x) is a subgroup of g.Let F be the filter generated by the subgroups fixg(E) for E E I. It is a normal

filter and so it defines a permutation model.Notice that x is symmetric if and only if there exists E E I such that

fix(E) ~ sym(x) .

Most cases we will examine are given by the ideal of finite subsets of A, thisgives the normal filter generated by the subgroups containing sets of the formfix(E) = {1T: 1T(a) = a for all a E E} for E c A finite.

Given such a model, if E is a finite set of atoms such that sym(x) ;;2 fix(E), wesay that E is a support of x. If fix(x) 2 fix(E) we say that E is a supersupportof X , and it means that not only E supports x, but it also supports each of itselements .

A fact which will be used later is that if we construct a permutation model inthe theory ZFA+AC, then AC holds in the kernel POO(O). In particular, every set

WEAK FORMS OF AC 51

in the kernel can be well-ordered. From this follows that for x in the permutationmodel, x can be well ordered if and only if there is a one-to-one mapping f of xinto the kernel. If f is such a mapping, 1rf = f if and only if 1r E fix(x) . Therefore,

V 1= "x can be well-ordered" if and only if fix(x) E F.

We describe now several of the better known permutation models.

The Basic Fraenkel Model (BFM):This model is obtained from a countable set A of atoms (A infinite), and the

group g of all permutations of A and the ideal I of finite subsets of A. In thismodel, the set A has no well-ordering and it is amorphous .

Mostowski's Ordered Model (MOM):Let A be a countable set and < be a dense linear ordering of A without end

points. (A is therefore isomorphic to the rationals.) Take g the group of all orderpreserving permutations of A and I the ideal of finite subsets of A.

In the resulting permutation model, A cannot be well-ordered, but there is alinear ordering of the universe. This shows that in ZFA, the Axiom of Choice isindependent from the Ordering Principle.

Fraenkel's Second Model (FSM):This model is designed to obtain the failure of the Axiom of Choice for families

of pairs. In this case, A is a countable set divided into countably many disjointpairs, A = U{Pn: nEw}, where for each nEw, Pn = {an,bn}.

The model is obtained using the group g of permutations of A which preservethe pairs Pn (in other words, the permutations 1r such that 1r({an, bn}) = {an, bn}) ,and the ideal I of finite subsets of A.

Each Pn and the sequence (Pn : nEw) are in the permutation model thusobtained. But in the model there is no function on w picking one element fromeach of the pairs Pn . This shows that ZFA does not prove C2 •

Generalizations of FSM (MFC(k)) :These are models obtained organizing the countable set A of atoms in disjoint

subsets of size k for some k E w.Define MFC(k), the model of finite cycles of size k, considering

where S, = {at, . . . ,ai} are pairwise disjoint k-element sets.For each i E w consider the permutation 1ri of A acting on S, by

and fixing all atoms outside of Si. The group g is generated by the set {1ri : i E w};therefore, g is the weak product of w copies of the cyclic group of order k. Withthis group g and the normal ideal I of finite subsets of A we obtain the modelMFC(k).

In MFC(k), Ck fails, but PP holds. Moreover in this model every infinite setcontains a countable collection of pairwise disjoint finite subsets . This fact is dueto Pincus, and we reproduce the proof here.

Lemma 2.1. For every k E w, MFC(k) satisfies that every infinite set contains acountable collection of pairwise disjoint finite subsets. Therefore, PP holds in thismodel.

Proof. Let X be infinite and let E be a support for X . Given x E X, let O(x) ={1I"X : x E fix(E)}, the orbit of x with respect to fix(E). Since fix(E) c sym(X),O(x) eX; we will see that each of these orbits is finite.

Since 11"2"1 011"1 E sym(x) implies 1I"1X = 1I"2X, it follows that O(x) has at most asmany elements as the quotient fix(E)/(fix(E)nsym(x)) . For a support Ex for x, wehave fix(E U Ex) c fix(E) n sym(x), and thus fix(E)/fix(E) n sym(x) has at mostas many elements as fix(E)/fix(EUEx), which in turn has as many elements as thegroup of permutations which only move elements of Ex <, E, since permutationsa,p E fix(E), do not move the elements of fix(E), and if they only differ on howthey move the elements outside of EUEx, they are equivalent modulo fix(EUEx)and correspond to the same element in the quotient. Therefore, since there is onlya finite number of permutations which only move elements in Ex <, E, the orbitO(x) is finite.

X is thus split in an infinite collection of finite sets . Moreover, the collection{O(x) : x E X} is well-orderable, since E is a supersupport for it ( for everya E fix(E) and every x E X, aO(x) = O(x)). We can thus take a countablecollection of orbits, and fix an enumeration. Taking the union of the even numberedorbits and collecting the rest of X together, we obtain a partition of X into twoinfinite subsets. D

A model with cycles of all prime lengths MFC(P):In this case also A = UiEW Si, but now each Si = {at, .. . , a~i} has Pi elements,

where Pi is the i th prime number . The model MFC(P) is constructed using finitesupports and the group of permutations 9 generated by the permutations 1I"i of Aacting in S, by

ii i i iat H a2 H a3 H . .. H api H a1

and fixing all atoms outside of Si .This model allows us to show that VnC n does not imply a result of ACF,

which is a result of [16].

A Model with Countable Supports (CSM ):It will be useful to consider the permutation model obtained starting from an

uncountable set of atoms A, with the group 9 of all permutations of A and theideal I of subsets of A which are at most countable. Clearly, this is a normal ideal.

In this model, AC fails, but every infinite set has a countable subset. Moreover,we have the following facts.

Lemma 2.2. For each n ~ 2, the choice principle Cn fails in CSM

WEAK FORMS OF AC 53

Proof. It is enough to consider the collection F = [A]n . If f is a selector for F,and E is a support for f, take a subset X of A <, E with n elements and let 1r bea permutation interchanging f(X) and any other element of X. A contradictionis reached in the usual way since 1r E fix(E) . 0

As a consequence, ACF, ACWO, OP, SP, and of course, AC , all fail inCSM.

Lemma 2.3. In CSM every infinite collection of non-empty sets has an infinitesubfamily with a choice function . So, every infin ite set contains an infinite countable subset.

Proof. Let F be an infinite collection of non-empty sets. Take, in the base model,a countable subcollection F ' C F and a choice function f for F'.

Choose a support Enfor each X n E F', and a support E~ for each f(Xn ) ,

X; E F' . Then, E = UnEw En U UnEwE~ is (still in the base model) a countablesubset of A. If 1r E fix(E) then n] = f . It follows that E is a support for t, andtherefore f is in the permutation model.

The second part of the Lemma is proved considering the family {[Xjn : n EW, n > O} for an infinite set X. 0

2.2. SYMMETRIC MODELS

We assume that the reader is familiar with the basic ideas of Boolean valuedmodels and generic extensions .

Symmetric submodels of generic extensions are obtained using ideas analogousto the ones described for permutation models. The difference is that in this casewe will start from a Boolean valued model and use a group of automorphisms ofthe Boolean algebra and a normal filter of subgroups to determine which elementsof the generic extension will belong to the symmetric model.

An automorphism tt of the Boolean algebra B is a bijection from B onto itselfwhich satisfies

1r(U + v) = 1rU + 1rV

1r(u·v) = 1rU '1rV

1r(-v) = -1rV

Clearly, an automorphism 1r of B preserves the partial ordering of B, and therefore,if B is complete, tt preserves infinite sums and products.

Each automorphism 1r of B can be extended by recursion to an E-automorphismof M B as follows:

1. 1r0 := 0;2. assuming 1rY has been defined for all y E dom(x) , define 1rX by

dom(1rx) .- 1r"dom(x) = {1rY : y E dom(x)}

1rx(1rY) .- 1r(x(y))

By induction it can be easily verified that 1rX = x for every x E M . The nextlemma is also proved by induction.

Lemma 2.4. Let cp(Xl,' .. ,xn ) be a formula with variables in M B, and let 1r anautomorphism of B. Then

Given a group g of automorphisms of B, and a normal filter:F on g, an elementx of the Boolean model is symmetric if the subgroup of g of automorphisms whichleave x fixed is in the normal filter. If G is a generic filter, the symmetric submodel of M[G] given by g and :F is the class {ia(x) : x hereditarily symmetric }of interpretations respect to G of hereditarily symmetric names. The symmetricmodel is a model of ZF .

The Basic Cohen model (BCM):This is the symmetric model obtained using the the complete Boolean algebra

of regular open sets of the partial ordering

P = {p : (p is a finite function) 1\ (dom(p) c w x w) 1\ (range(p) C {a, I})}

which adds countably many generic reals (the order relation is given by the extension relation between functions) .

The group is that of all automorphisms of the Boolean algebra induced bythose automorphisms of the partial ordering which produce a permutation of thegeneric reals. The filter is the normal filter of finite sets.

In this model the set of real numbers cannot be well ordered. Therefore, ACcannot be proved in ZF .

Halpern and Levy [10] established several properties of BCM. In particularthey showed that The Selection Principle SP holds in this model.

It is useful to remember that SP is equivalent to the following statement ([13]) :

For every set M there is an ordinal a and a one-to-one mapping of M into thepower set of a .

From this follows that SP implies OP, since the lexicographic ordering linearlyorders P(a).

To see that SP holds in BCM, Halpern and Levy showed that in BCM there isa one-to-one mapping of the universe into [A]<wx On, where A is the countable setof generic reals. So if M is any set in the symmetric model, there must be an ordinala and a one-to-one function f : M -t [A]<w x a . For x E M, if f(x) = (E,[3),define g(x) = {{3 + n : nEE}. The function 9 : M -t P(w. a) is one-to-one andthe principle of Kinna and Wagner holds.

3. Restricted and Diminished Choice Principles

The restricted and diminished choice principles presented in this section are weakforms of choice obtained by debilitating in certain specific ways some of the usualchoice principles (see [4]) .

WEAK FORMS OF AC 55

The Axiom of Choice AC can be stated saying that for every set X, there is achoice function for the family of all non-empty subsets of X.

If we require the choice function to exist only on the collection of non-emptysubsets of an infinite subset of X, we obtain the restricted version of AC denotedby

ACO: For every infinite X there is an infinite Y ~ X and a choice function forthe collection of non-empty subsets of Y.

Notice that AC can also be written as VX(X is well orderable); the restrictedversion of this statement is:

('IX infinite )(3Y C X infinite )(Y is well orderable) .

It is easy to see that both restricted principles turn out to be equivalent to thestatement" Every infinite set contains an infinite countable subset" , which in turnis equivalent to "every Dedekind-finite set is finite".

For ACF we can do something similar. If we express ACF as 'IX there is achoice function for the collection of finite non-empty subsets of X, its restrictedversion is

ACFO: VX3Y ~ X, such that Y is infinite and there is a choice function for thecollection of finite non-empty subsets of Y.

Consider the following additional "restricted" choice principles.

SpO: For every infinite set X there is an infinite subset Y and a function assigninga proper subset to every subset ofY with at least two elements.

Opo: For every infinite set X there is an infinite subset Y and a linear order forY.

ACWOO: For every infinite set X there is an infinite subset Y and a functionassigning one oEits elements to every non-empty well orderable subset oEY.

C n 0: For every infinite set X there is an infinite subset Y and a function choosingone element from each n-element subset ofY.

ACWOOis equivalent to ACFo . One implication is obvious; for the other,for X infinite, if it has a countable infinite subset Y, then an enumeration ofY provides a well order for every nonempty subset of Y (and therefore a choicefunction for the collection of such subsets of Y); if there is no such subset Y, thenthe only well orderable subsets of X are the finite ones.

The restricted version of a choice principle depends on the specific way theoriginal principle is stated. Restricted versions of two equivalent choice principlesmight give non-equivalent principles. So we take the list above as the definitionsof our restricted choice principles.

We define now the "diminished" choice principles.If we state AC asFor every family of non-empty sets there is a choice function,


its diminished version is

AC-: For every infinite family F ofnon-empty sets, there is an infinite subfamilywith a choice function.

This is different from the restricted version stated above; in fact, this diminishedversion of AC is equivalent to ACw [2].

Other "diminished" principles we will consider are the following.

SP-: for every infinite family F ofnon-empty sets with at least two elements each,there is an infinite subfamily F' C F and a function assigning a proper subsetto each element of F'.

ACWO-: for all infinite family F of non-empty well orderable sets, there is aninfinite subfamily F' C F with a choice function.

ACF-: for all infinite family F of finite non-empty sets, there is an infinite subfamily F' C F with a choice function .".

C n -: for every infinite family F of finite non-empty sets ofn elements, there is aninfinite subfamily F' C F with a choice function.

Each of the diminished and restricted principles follows from the correspondingchoice principle. The implications are obvious, and the fact that they are strictcan be shown using the permutation model CSM of countable supports (see 2.2and 2.3) .

The relationship between restricted and diminished versions is interesting. ForAC, the diminished version strictly implies the restricted version [i.e, ACw strictlyimplies "every infinite set contains a countable subset"), whereas for ACF a strictimplication goes in the opposite direction. For ACWO and for SP neither implication holds. Some of these facts will be proved in section 5.

Theorem 3.1. (See lsl-)1. AC- strictly implies ACOE. ACFoimplies strictly ACF-3. ACWOo does not imply nor is implied by ACWO- .4. Spo does not imply nor is implied by SP-.

The finite choice principles Cn, C n0 , and C n- pose interesting questions. Forexample, C2 implies C4 , but it does not imply C3 • A complete characterization ofthe implication relationships between these choice principles for families of finitesets appeared in [7]. Gauntt gives a set D of conditions to be satisfied by a finiteset E of natural numbers and a number m for the following equivalence to hold.

[(Vn E Z)Cn implies Cm ] iff D(Z, m).

The following result of [3] gives a useful criterion for implications between finitechoice principles. The proof uses the models MFC(k).

Theorem 3.2.a) If k divides m, then Cm implies Ck, and Cm - implies Ck -.b) If there is a prime number p which divides k but does not divide m, then

Cm does not imply Ck - (in ZFA).

WEAK FORMS OF AC 57

For n = 2,3, C n 0 strictly implies C n-. Montenegro [17] has shown that theimplication is true for n = 4; but it is not known to hold for n 2: 5. Note thatfor every nEw, Cn - holds in the Basic Fraenkel Model but Cn

0 does not . It isobvious that (VnCn)O implies (VnCn0), but it is not clear if this implication isstrict.

The principle ACF- implies that the union of countably many finite sets iscountable.

4. Another Permutation Model

We describe now another permutation model, the Cylinder Model (CM), in whichthe set of atoms is organized in a countable collection of countable dense "circles"which can rotate rigidly (see [4]).

More formally; y/e consider the additive group R l = Q/Z (isomorphic to thesubgroup of 8 1 formed by the points with rational argument) . To define the groupg of permutations consider A organized as follows

A={af :iEw,pERl}

where af =aJ implies p =q and i =j .A is thus the union of a countable collection of copies of R l ; we define Ai =

{af : pERl}, for each i E w.Define g as the group generated by {1I"f : i E w,p E R l } , where

1I"P(aq) = ap+q

t t t

andP( q) q if ' '''/' .1I"i aj = aj , 1 J r z.

The Cylinder Model (CM) is obtained using this group and the ideal of I offinite sets.Lemma 4.1. For each n 2: 2, the family [A]n does not have a choice function inCM.

Proof. [A]n is clearly in the model. Suppose that f is a selector for the family'of n-element sets of atoms, and let E be a support for f . Let k E w be suchthat Ak n E = 0; and consider the permutation 11" = 1I"~1/nl, where [r] is the cosetcorresponding to the rational r. If Y = {a~l, a~l/n], a~/nJ, .. . ,a~n-l/nl} (Y is theset of vertices of a regular polygon of n vertices), then,

",y = ({ [0] [lin] [2/n] [(n-l)/n]})II 11" ak , ak , ak , • •• , ak

{[O+(l/n)] [(l/n)+(l/n)] [(2/n)+(1/n)] [(n-l)/n+(l/n)]}= ak , ak ' ak , . . . , ak

= {[l/n] [2/n] [3/n ] [OJ}ak ,ak ,a k , •. • ,ak

= y

and11" E fix(E) c sym(f)

So, (Y,f(Y)) E f implies 1I"(Y,f(Y)) E 1I"f. It follows that (Y,1I"(J(Y))) E f, since11" E sym(Y) and 11" E sym(J). But f is a function, so 1I"(J(Y)) = f(Y)j since f is achoice function, f(Y) E Y, but no element ofY is fixed under 11" . Contradiction. 0

Lemma 4.2. CM satisfies "Every infinite set contains a countable subset" .

Proof. The proof follows ideas of [1].For X E CM infinite, we need only consider the case in which X is not well

orderable. By [Jel , (4.2) p. 47J there is no finite supersupport E for X . So if wefix a support Eo for X, we have that

fix(X) 1> fix(Eo)

and therefore there are 11" E fix(Eo) and Xo E X such that 11" f/. sym(xo). Fix alsosuch xo, and observe that

sym(xo) 1> fix(Eo) (2)

IT E is a support for Xo, then, by (2), E et Eo, we pick a support E l for Xo suchthat El "Eo has the least possible number of elements. Fix a E E l "Eo, and letF =E l <, {a}. Let k be such that a E Ak, define now a function f : Ak -t X by:

f = ((1I"a,1I"xo) : 11" E fix(F U Eo)}

As in [1], f is a function in CM, its domain is Ak and f"Ak eX.To complete the proof we only need to show that I"Ak is infinite.Let h : Rl -t fix(F U Eo) be defined by h(P) = 1I"f. Notice range(h) C fix(F U

Eo), since (F U Eo) n Ak = OJ Clearly h is a monomorphism, and G = {p E R l :

h(P)(xo) =xo} is a subgroup of Rl . Each element of Rl fG gives rise to a differentelement of f" Ak, since if p, q E Rl are such that p - q f/. G, then

h(p - q)(xo) f. Xo ·

That is to say, if p - q f/. G, then the followingstring of non-equalities shows thatp and q give rise to different elements of f" Ak:

h(q)-lh(p)(xo) f. Xo

h(P)(xo) f. h(q)(xo)

1I"fxo f. 1I"Z XO

f(1I"fao) f. f(1I"Z ao)

We also have that Rl f. G, since otherwise we contradict the minimality of E l .

So the following lemma is enough to get that f" Ak is infinite.

Lemma 4.3. There is no proper subgroup G of R l such that R l fG is finite .

WEAK FORMS OF AC 59

We have then, f : Ak ~ X with f"Ak infinite, and Ak countable (in fact,fix(Ak) :) fix( {a}). Therefore {a} supports the bijection p t-+ at from R1 onto Ak,since 1r E fix({a}) and

1r(p,an = (1rp ,1rat)

= (p,at)

since p is in the kernel of CM.Given a bijection 9 : w ~ Ak, it easy to find a countable infinite subset of X.

o

Theorem 4.4. ACO does not imply C n for any n ~ 2

Proof. From the previous lemmas it follows that CM satisfies

o

D. Pincus has constructed a permutation model where ACoholds and there isa countable family of countable sets without a choice function ([19], see also [11],p.132 problem 4), this is also the case in the cylinder model. Pincus' model hasa countable collection of countable sets of atoms, with the group of permutationsgenerated by the permutations of each of these sets.

This provides another way to see that none of the restricted versions of thechoice principles we have considered implies its original version, because all therestricted versions are implied by ACo, but each of the original versions impliessome c; The principles SpO, OpO, ACFO, (VnCn)O, (VnCn0), each c;0, andPP all follow immediately from ACO.

Observe also that in CM, each Ai is well ordered since the rationals can be wellordered in this model. Thus, {Ai : i E w} is a family of well ordered sets withoutan infinite subcollection with a selector. So, ACWO- and AC- are false in CM,which shows that neither of these principles is deducible from ACo .

5. Some Implications and Independence Results

The models described in the previous sections are useful to prove some independence results regarding choice principles. We will give some indications about howto prove some implications and some independence results.

Part 1 of Theorem 3.1 is well known, we include a proof for completeness.

Lemma 5.1. AC-implies ACO.

Proof. let X be an infinite set, and consider the family F = {(x)n : nEw} (where(X)" is the set of one-one sequences of length n of elements of X). Using AC- weget a choice function f for an infinite subset F' of F . The union of the ranges ofall the chosen sequences is a countable infinite subset of X . 0

60 DE LA CRUZ AND DI PRlSCO

The fact that ACOdoes not imply AC-can be obtained using the cylindermodel. As we have seen, ACOholds in this model, but AC-does not.

Lemma 5.2. ACFO implies ACF- .

Proof. Given a family F of pairwise disjoint finite sets, ACFOimplies that thereis an infinite Y C UF with a choice function f defined on the set of its finitesubsets. Clearly G = {a E F : anY ::f. O} is infinite, and so we define a choicefunction on G as follows:

g(a) = f(a n Y) for all a E G.

oTruss proved in [26] that for any n > 1, Cn implies that there are no amorphous

sets, the same proof works if we assume C no instead of Cn' Therefore C nc is nottrue in BFM for any n (nor ACFO, in consequence) . In conclusion, Fraenkel'sBasic Model provides a model of ZFA where ACF- holds but ACFo does not.

The next lemma is useful to prove that ACWO- holds in the Basic FraenkelModel. Notice that ACWOo does not hold in BFM since it implies that no set isamorphous .

Lemma 5.3. In the Basic Fraenkel Model every well orderable set has a minimumsupersupport, and the (class) function £ that sends every well orderable set to itsminimal supersupport (as a class in the model) is hereditarily symmetric.

Proof. The existence of a supersupport for every well orderable set follows from[11, (4.2) p. 47]. Standard arguments can be used to show that the intersection oftwo supersupports is a supersupport and that £ is in the model. 0

Proposition 5.4. (See Uj.J ACWO- holds in BFM.

With a similar kind of argument we can get the following.

Proposition 5.5. SP- holds in BFM and in MOM.

Nevertheless, AC-is false in both BFM and MOM.

Pincus's Model mentioned after Theorem 4.4 allows us to show that ACWOdoes not imply SP-. If a countable set of atoms is organized as a countablesequence {An : nEw} of infinite sets, the permutation model obtained with thegroup generated by the permutations of each set An, with finite supports, satisfiesACWO- (since ACo holds there) but not SP-.

Proposition 5.6. ACWOO is false in BFM.

Proof. Clearly ACWOO implies C20 , and by the results of [26] mentioned above,

this implies that there are no amorphous sets. Since A is amorphous, we get thedesired result. 0

Proposition 5.7. AC-is false in BFM.

Proof. By 3.1 and the fact that ACodoes not hold in BFM. o

WEAK FORMS OF AC 61

In the cylinder model, ACWOo holds but ACWO- does not .For the Selection Principle, none of the implications hold. The cylinder model

satisfies SpOand ,SP-, and BFM satisfies SP-and ,SPO.

6. Ramsey's Theorem, Partitions and Weak Choice Principles

RAMSEY'S THEOREM (RT): for every partition of the collection of n-element subsets of an infinite set A into k pieces, there is a infinite subset H ~ A withall of its n-element subsets in the same piece of the partition.

In other words, Ramsey's Theorem asserts for every infinite set A and everyF : [A]n --+ k, there is H ~ A, H infinite, and i < k such that F"[H]n = {i} (see[22]) . The set H is said to be homogeneous for the partition F.

If A is a well-ordered set , then the theorem can be proved in ZF. For arbitrarysets, some choice is necessary. In fact, RT implies ACF-: Given a family F offinite sets, let A = uF and partition [A]2 into two pieces putting F( {a,b}) = 0 ifand only if a and b do not belong to the same element of F . If H is an infinitehomogeneous set for this partition, either it is contained in a single element of F ,which is impossible since these elements are finite; or it is a selector for an infinitesubfamily of F . (See [15] .)

On the other hand, RT follows from ACO, since once we have a countablesubset B of the infinite set A, the fact that B is well ordered, allows us to carryon the usual proof of the theorem starting from the set B.

Blass studies in [1] the relationship between RT and weak choice principles.His results, together with some facts which were known already, give the following.

The Boolean Prime Ideal Theorem, SP, the order extension principle, ACWO,ACF, C 2 , do not imply RT (not even jointly) and are not implied by RT.

The principles AC, 'VK,AC"" dependent choice of any infinite length K, (DC",),nonexistence of infinite Dedekind-finite sets, all imply RT, but none of them followsfrom RT.

The independence results are proved using the next theorem.

Theorem 6.1. ([lJ, see also [14J.) Ramsey's Theorem holds in the Basic FraenkelModel, but it is false in the Basic Cohen Model.

Proof. (Sketch) Lemma 4.2 states that ACo holds in CM. The proof of Lemma4.2 follows the lines of [1], where it is shown that ACo holds in BFM. Since RTis implied by ACO, it holds in both models .

To show that RT does not hold in BCM, Blass uses some results regardingthe structure of the reals in this model. Namely, that the set A of generic reals isdense in 2w , and that if Y ~ A is infinite , there is a basic neighborhood U (in theproduct topology of 2W

) such that UnA ~ Y . With these facts, a counterexampleto Ramsey's Theorem can be given as follows. Define a partition f : [A]2 --+ 2 byf(x, y) = 0 if and only if the least member of the symmetric difference xtly is~. 0

Since the Hahn-Banach Theorem and the existence of an ultrafilter on w followfrom the Boolean Prime Ideal Theorem, they do not (even jointly) imply RT.

62 DE LA CRUZ AND Dr PRISCO

RT holds in Solovay's model where all sets of real numbers are Lebesgue measurable, since it follows from the Axiom of Dependent Choices which holds in themodel. But both the Hahn-Banach Theorem and the existence of an ultrafilter onw fail in this model([25]). Thus, RT does not imply any of them.

Some related results are presented in [4]: Ramsey's Theorem holds in CM but,as we have observed, ACWO- does not hold there; this establishes that ACFodoes not follow from RT. On the other hand, Ramsey's Theorem is not provablefrom SpO (which is, in our scheme, the next weaker principle after ACO ).We willshow this below using Cohen's basic symmetric model.

The existence of partitions of infinite sets in the absence of AC is closely relatedto weak choice principles. We review some recent results in this respect .

If an infinite set is not amorphous, we say it is partible (it can be split intotwo infinite subsets).

In [8] , Gonzalez analyzes a series of properties arising from the possibility ofsplitting infinite sets. We have already mentioned that "every infinite set can besplit in two infinite subsets" is not provable in ZF (it is false in BFM and this canbe transfered to a model of ZF).

A set is n-partible if it can be split into n infinite subsets, and it is < w-partibleif it is n-partible for every nEw, n > O. We say A is No-partible if it can be splitinto a countable sequence of subsets. A set is co-partible if it can be split intoinfinitely many infinite subsets.

A set A is said to be T-infinite if there is an infinite ~-chain in P(A) . Noticethat every linearly ordered set is T -infinite.

Theorem 6.2. The following implications hold for an infinite set A:A is No-partible '* A is co-partible '* A is < us-partible '* A is n-partible for

each n > 0 '* A is partible .Also, A is No-partible '* A is T -infinite '* A is < w-partible.None of the implications is reversible, and the principles co-partible and A is

T -infinite do not imply each other.

Gonzalez also shows in [8] that all these properties are notions of infinity (in thesense of Degen [5]) intermediate between the properties "A is Dedekind-infinite"and "A is infinite".

For the benefit of the reader we recall the definition.

Definition. (See [5] .) A notion of infinity for ZF is a formula ¢(x) with exactlyone free variable x such that the following are theorems of ZF

i) Va ~ w¢(a),ii) "In < w(..,</>(n)),iii) VxVy(lxl = Iyl&¢(x) '* ¢(y)),iv) VxVy(x ~ y&¢(x) '* ¢(y)) .

We say that two notions of infinity ¢ and t/J are incomparable if there is a modelN and sets u, v E N such that (in N) ¢(x) and ..,t/J(x) and t/J(y) and ..,¢(y) .

Gonzalez also solves a question of Degen ([5] , page 124) by showing that thereare incomparable notions of infinity. For this, he constructs a permutation model ,which is a pasting of BFM and MOM. Namely starting from two disjoint countable

WEAK FORMS OF AC 63

sets of atoms A and B, and a dense linear ordering < without endpoints of B, thepermutation model is obtained using the group 9 of all permutations 7r such that

i) 7r is a permutation of A and a permutation of B ,andii) 7r preserves -c .

In the model obtained, A x A is co-partible, but not T-infinite, and B is T-infinite,but not co-partible.

Ramsey's Theorem does not imply C2-, nor does it imply "every set is partible" (PP), since in the Basic Fraenkel Model, Ramsey's Theorem holds [1], butthe set of atoms is not partible.

In the basic Cohen model, the selection principle holds, but, as we have mentioned, Ramsey's Theorem does not . Thus, a consequence of the following lemmais that the principle "Every set is No-partible" does not imply Ramsey's Theorem.

Lemma 6.3. The Selection Principle implies every infinite set is No -partible

Proof. Given an infinite set X, let f : P>2(X) -7 P(X) be a selection function(i.e. for every S ~ X with at least two elements, 0~ f(S) ~ S).

From f we will obtain an w-sequence Ao,AI, . .. of distinct subsets of X, fromwhich follows that X is No-partible.

If X - f(X) is infinite, put Ao = f(X), otherwise Ao = X - f(X) . If we havedefined An, let Bn =Uf=oA i and put An+! = f(X -Bn) if Xf(X -Bn) is infinite,otherwise, put An+! = Xf(X - Bn ) . 0

Corollary 6.4. Every infinite set is No -partible does not imply Ramsey's Theorem.

Gonzalez [8] and Pincus [21] have obtained results regarding the principle DO:Every infinite set has a dense linear ordering.

Theorem 6.5. The following implications are provable in ZF:

AC~SP~DO~ OPj

and none of the implications is reversible in ZF+ the Prime Ideal Principle.

The first implication and its irreversibility is in [10] and the third in [8]. Theproof of the irreversibility ofSP~ DO is due to Pincus and has led him to considersome colored versions of Ramsey's Theorem and colored Ehrenfeucht-MostowskiTheorems. His proof uses a permutation model similar to MOM, but in which theatoms are endowed with a dense ordering and countably many densely dispersedcolors.

7. Transfer to ZF

As we mentioned above, the independence results obtained using permutationmodels are theorems in ZFA. To obtain corresponding independence results in ZFtransfer theorems of Jech-Sochor [11] and of Pincus [20] can be used.

The results of Jech and Sochor imply the following.


Proposition 7.1. If 1/J is a formula of the form 3Xcp(X, 13) , where the only quantifications in cp are of the types 3u E pIJ(X) and Vu E pIJ(X), and W is apermutation model such that W F= 1/Ji then there is a symmetric model N of ZFsuch that N F= 1/J.

Formulas like 1/J above are called boundable. Proposition 7.1 is useful for transfering negations of choice principles.

For example, for the Selection Principle we have that ...,SpO can be written as

3AVx E PW(A)VA' E PW(A)(A' C A&A infinite ~x is not a selector for P~2(A')).

Therefore, since ...,SpO holds in Mostowski's ordered model, it holds in a symmetric model of ZF.

The negation of "every set is partible" is the sentence ...,PP:

3X(X infinite &W C XVZ c X(Y infinite &Z infinite ~ Y n Z i= 0)).

Notice that ...,PP is boundable since "u is infinite" can be expressed by"In E PW+W(X)«n E w)& (J : n ~ u) ~ (J is not a bijection)) .

So, the existence of an amorphous set can be transfered from permutation modelsto symmetric models of ZF.

To deal with transfer of the choice principles themselves , we can use the resultsof Pincus. First, some definitions.

Definition. Given a set x, its injective cardinal isIxl- = sup{lo:l : there is an injection from 0: into z},

Definition. A term u(x) of the language of set theory with atoms is said to beboundable if there is an ordinal 13 (possibly depending on x) such that

Definition. A sentence of the form 3y~(y) is said to be injectively boundable if~(y) is a finite conjunction offormulas of the form

Vx[(lxl_ ~ u(y) 1\ x n trcl(y) = 0) ~ '1'(x, V)], (1)

where u(x) is a boundable term and '1'(x,y) is a boundable formula .

Theorem 7.2 (Pincus). If a sentence T is injectively bounded and it holds in apermutation model W , then T is consistent with ZF .

We will see that some of the choice principles considered can be shown to beinjectively bounded.

For example, ACo is injectively boundable. First, notice that Ixl- ~ w impliesthat w cannot be injected in x, and therefore x does not contain a countable infinitesubset. So, we express ACO by

Vx(lxl_ ~ w ~ (x is finite),

WEAK FORMS OF AC

and "x is finite" is boundable since it is expressed as

(3n E PW(O))(x is equipotent with n).

65

Therefore, ACo becomes the formula3yVx((lxl_ $ W & x n trcl(y) = 0) -t (3n E PW(O)(x is equipotent with n))).RT is also injectively boundable. We will write RT(X) to mean "for every

partition of [X]" in m pieces, there is HeX, H infinite, such that [H]n is containedin one of the pieces".

Thus, RT is the sentence VX(X infinite -t RT(X)). If X has a countableinfinite subset, then RT(X) holds, therefore RT is the same as

VX(X infinite without countable infinite subset -t RT(X)),

and this is equivalent to

3yVx ((Ixl- $ w&x n trcl(y) =0) -t (RT(X))).

Finally, we show that ACWO-is injectively boundable. Write ACWO- (F)to express

(F infinite &0 ~ F 1\ Vx E F x is well orderable) -t 3F' c F (F' infinite & 3/(f is a choice function for F')).

Now, ACWO- is equivalent to

VF(IFI- -t ACWO-(F)). (2)

Clearly ACWO-implies (1), and assuming (1), if F is an infinite family of wellorderable non-empty sets, we consider two possibilities. If F does not have a countable subset, IFI- = W $ WI, and by (1), ACWO-(F) follows. Otherwise, F hasan infinite countable subset F', and IF'I- =WI, thus by (1), ACWO-(F').

With these observations we are able to transfer the results of the previoussections to ZF. For example, to show that in ZF ACo does not imply AC- , it isenough to recall that the conjunction ACO 1\ ..,AC- holds in the Cylinder Model,and as each conjunct is either boundable or can be written in the form (1), theconjunction is injectively bounded and therefore, consistent with ZF.

Figures 1 to 6 summarize the results concerning some of the principles andmodels mentioned so far. For each diagram, the principles in the shaded regionare true in the corresponding model, and the others are false. Finally, Figure 7contains most of the implication and independence results (in ZF) mentioned inthis paper.

66 DE LA CRUZ AND Dr PRlSCO

I AC- I1 sp- I!

II RT f:

[xcwo-]

IACF-I

I IVnC;; II I c; I

I ACo II spo II opo IIACWo ol

I ACFo I

I(Vn Cn)OI

IVn(Cn0)

I Cmo

I pp I

I Cm I

AC Isp I

I OP IIACWO II ACF II VnCn I

Figure 1. Choice Principles that hold in BFM

I AC- II ACo IAC III sp - II SP I I sr- I

II OP I or- II RT I~ ACWO I ~ACWOo ~ IACWO-1

II ACF I I ACFo I IACF- I

I VnC: I j (VnCn)OI

IVn(Cn0)I j Vn C;; III c; I I o.: I I C- Im

I PP IFigure 2. Choice Pr inciples that hold in MOM

WEAK FORMS OF AC 67

I AC- II SP- I

,.-------"

IACWO-I

I ACF-I

ACO ISpo I

I Opo II RT

IACWool

I ACFo I

SP

AC

OP

IACWO

I ACF

I vn~n I(VnCnt'l

IVn(CnO) I 1 VnC;; 1

I Cm I I Cmo I IC-

Im

I pp I

Figure 3. Choice Principles that hold in MCF(P)

AC

SP

OP

IACWO

I ACF

I VnCn

I Cm I

- -I ACo t I AC- II sr- I I SP- I

II or- ~ RT ~

IACWool IACWo -1

;1 ACFo ~ I ACF- II(Vn Cn)OI

IVn(Cn0)I 1VnC;; II ,cmo I I c; ~

I PP IFigure 4. Choice Principles that hold in CM

68 DE LA CRUZ AND Dr PRlSCO

AC ISP I

I OP IIACWO II ACF II VnCn I

I Cm I

I ACo I I AC- II sr- I I SP - I,

I or- II RTE

[xcwo-] ~ACWO - ~

II ACFO1 'I ACF- IIr(VnCn)Ol

IfVn(Cn0)1 :1 VnC;; I

II c mo 1 ,1 c; t

If PP -1

Figure 5. Choice Principles that hold in CSM

AC

SP

OP

ACWO

ACF

AC-

Figure 6. Choice Principles that hold in BCM

WEAK FORMS OF AC 69

Figure 7. Implications between Choice Principles

70

References

DE LA CRUZ AND DI PRISCO

1. Blass, A., Ramsey's Theorem in the hierarchy of choice principles, Journal ofSymbolic Logic 42 (1977) 367-390

2. Brunner, N., Sequential compactness and the axiom of choice, Notre Dame Journalof Formal Logic 24 (1983) 89-92

3. De la Cruz, 0 ., Relaciones de implicaci6n e independencia entre principios debilesde elecci6n, M.Sc. Thesis, Instituto Venezolano de Investigaciones Cientificas, 1995.

4. De la Cruz, O. and C.A. Di Prisco, Weak Choice Principles, Proceedings of theAmer. Math. Soc. to appear.

5. Degen, J.W. , Some aspects and examples of infinity notions, Math. Logic Quarterly40 (1994) 11-124.

6. Feigner, U. and J. Truss, The independence of the prime ideal theorem from theorder-extension principle, Preprint (1996) .

7. Gauntt, R. J., Axiom of choice for finite sets - A solution to a problem of Mostowski,Notices Amer. Math. Soc. 11 (1970) 454.

8. Gonzalez, C.G., Ordens densas, partic;Oes e 0 axioma da escolha, in: Tese deDoutorado, Universidade Estadual de Campinas, Brasil, 1994.

9. , Dense orderings, partitions and weak forms of choice, Fund. Math. 141(1995) 11-25.

10. Halpern D. and A. Levy, The ordering theorem does not imply the axiom of choice,Notices Amer. Math. Soc. 11 (1964) 56.

11. Jech, T., The Axiom of Choice, North Holland, Amsterdam, 1973.12. Jech , T., Set Theory, Academic Press, New York, 1978.13. Kinna, W. and K. Wagner, Uber eine Abschwiinchung des Auswahlpostulates,

Fund. Math . 42 (1955) 75-82.14. Kleinberg, E.M., The independence of Ramsey's Theorem, Journal of Symbolic

Logic 34 (1969) 205-206.15. , Infinitary Combinatorics, in: Cambridge Summer School in Mathematical

Logic, (A.R.D. Mathias, ed.), Lecture Notes in Math., 331 (1973) 361-418.16. Levy, A., Axioms of multiple choice, F\md. Math. 50 (1962) 475-483.17. Montenegro, C., Weak versions of the axiom of choice for families of finite sets , in:

X Simposio Latinoamericano de L6gica Matematica, to appear.18. Moore, G.H., Zertnela 's Axiom of Choice. Its origins, development, and influence

Springer Verlag, Berlin, 1982.19. Pincus, D., Individuals in Zermelo-Fraenkel Set Theory, Doctoral Dissertation,

Harvard University, 1969.20. , Zermelo-Fraenkel Consistency Results by Fraenkel-Mostowski Methods,

Journal of Symbolic Logic 31 (1972) 721-724.21. , The dense linear ordering principle, Journal of Symbolic Logic 62 (1997)

438-456.22. Ramsey, F.P., On a problem of formal logic, Proc. of the London Mathematical

Society, Ser . 2 30 (4) (1928) 338-38423. Rubin, H. and J. Rubin, Equivalents of the axiom of choice, North Holland, Ams

terdam, 1963.24. , Equivalents of the axiom of choice, II, North Holland, Amsterdam, 1985.25. Solovay, R.M., A model of set theory in which every set of reals is Lebesgue

measurable, Annals of Mathematics 92 (1970) 1-56.26. Truss , J., Classes of Dedekind finite cardinals , Fundamenta Mathematicae

LXXXIV (1974) 187-208.27. , Permutations and the axiom of choice, in: Automorphisms of first order

structures, (R. Kaye and D. Macpherson, eds.), Oxford Univ. Press, (1994) 131-152.28. , The structure of amorphous sets, Annals of Pure and Applied Logic 13

(1995) 191-233.

A TASTE OF PROPER FORCING

MARTIN GOLDSTERN1

Technische UniversitiitWiedner Hauptstrafte 8/118A-l040 WienAustria

Abstract. We review basic definitions and theorems in Shelah's theory of ProperForcing.

The contents of this paper correspond roughly to a series of talks that I gaveat the "Set Theory and Applications" workshop held in June 1996at the CRM inBarcelona, Spain.

Since those lectures as well as these notes were meant to be an introductorycourse on proper forcing, there are no new mathematical results here. I hope thatthese notes can

(a) serve as a "springboard" for diving into deeper literature (such as chaptersIII, X and XII of Shelah's books [9J and [10]2)

(b) inspire others to continue to expand this "secondary" or "talmudic" literatureon proper forcing

(c) popularize the (conscious) use of the "alphabet convention" 1.2.

1. Notation and Conventions

Claim 1.1. We use standard set theoretic notation, as it is found in the books ofJech [6J and Kunen [7] .

This claim is of course wrong.We use "upwards" notation for forcing, see below. However, we still call the

generic subset of P "filter", although technically speaking it is an ideal.For technical reasons all our forcing notions IP' come equipped with a weakest

element 0p •

II am grateful to the CRM in general and to Joan Bagaria and A.R.D. Mathias inparticular for inviting me to their conference and for relentlessly reminding me of mypromise to write up these notes.

20 ne of the main open problems in th is field is the question when this book willappear.

71

C.A. Di Prisco et al. (ed.), Set Theory, 71-82.© 1998 Kluwer Academic Publishers.

72 MARTIN GOLDSTERN

Names in a forcing language (or variables ranging over such names) are usually(but not always) marked by a tilde, as in g . Standard names for objects in theground model are in principle marked by a "check" accent (as in Ii), but we almostalways omit it. G or Gp is the canonical name for a generic filter, but often alsostands for a variable ranging over all V-generic filters.

Let g[G] denote the "evaluation" of the name g by the filter G (sometimeswritten valG(g))

We occasionally confuse the set of names, vP with an arbitrary generic extension V[Gp]. We usually assume that V is the whole universe, but occasionally treatit as a countable model over which we can find generic filters.

We also confuse a forcing notion lP' = (P,~ , 0p) with its carrier set P.If lP' is a forcing notion, q E P, and A ~ lP' (typically, a maximal antichain), we

write Afq be the set of elements of A "selected" by q, i.e., all a E A which arecompatible with q.

Also, dom(J) and ran(J) stand for the domain and range, respectively, of afunction (or relation) f.

QUO VADIS?

Traditionally, there are two (contradictory) notations for interpreting a partialorder as a forcing notion . A majority of set theorists (including the books by Kunenand Jech) uses the "Boolean" or "downwards" notation, where q ~ P means that qis "stronger" than P (and in particular, q ~ P~ q II-P E Gp), citing the universalagreement on the standard order of a boolean algebra or a lattice: A conjunctionp A q is traditionally considered to be smaller than its constituents.

The "Israeli" or "upwards" tradition (used not only by Shelah and some of hiscoauthors but also by Cohen in his original paper) expresses the same concept byq ~ p (arguing that q has "more" information than p) .

(A third possibility is to abandon ~ altogether and only use q II- p as anabbreviation for q II-pEG.)

We use here the Israeli notation, but to make it easier for the readers in the"Boolean" camp we in addition employ the alphabet convention, a notation whichis compatible with the upwards as well as with the downwards interpretation.

Definition 1.2. The alphabet conventionWhenever two conditions are comparable, the notation is chosen so that the

variable used for the stronger condition comes "lexicographically" later.

For example, we can have a condition q which is strictly stronger than p, butwe try to avoid the converse situation. Similarly, a condition called P2 or p~ isallowed to be either stronger than PI or incompatible with Pll but not (strictly)weaker.

Note that in some isolated cases the alphabet convention may be impossibleor inconvenient to execute, for example if we work with quasiorders and have toestablish q ~ P and P ~ q, or if conditions are denoted by expressions such as pAqor [cp(r,§)].

A TASTE OF PROPER FORCING 73

2. Proper Forcing

Imagine you want to construct a model of set theory in which 2No = N2 • A naturalway to do this is to construct a model V1 = vt, where Vo is the "ground model"satisfying GCH (this makes various combinatorics easier), and lP is a forcing notionadding N2 Vo many reals. This can of course only work if we can also ensure thatN2Vo =N2 VI • A popular way to achieve this equality is to use a forcing notion thatsatisfies the countable chain condition. The ccc is helpful because of the followingfact.

Fact 2.1. Let lP satisfy the ccc, Il-p "{1 ~ Ord is countable [or of cardinality ~ x]",then there is a set B ~ Ord which is countable [or of cardinality ~ ~] withII-p A C B .

In ~articular, II-p "N1V is uncountable and hence N1v = N1vP

" , since in VIP anycountable subset of N1v is covered by a countable set from V, hence is boundedbelow N1-.

However, in many cases ccc forcing is inappropriate. Properness is a propertyof forcing notions which is weaker than ccc (and at the same time also a weakeningof "a-closed") and is still sufficient for not collapsing Wl' Instead of 2.1, a typicalproper forcing construction will use the following fact (see 2.8 below):Fact 2.2. If lP is proper (defined below), then:

- whenever p II- "{1 ~ Ord is countable", then there is q ? p and a countableset B such that q II- {1 ~ B .

- If moreover lP satisfies the N2-cc (i.e., any antichain of lP has size at most N1) ,

then lP preserves all cardinalities and cofinalities.

We will give several equivalent definitions of properness :

Definition 2.3. Let lP be a forcing notion, pEP. The antichain game Gac(lP,p)is defined as follows: Player I plays a maximal antichain Ao above p. Player IIresponds with a countable (i.e., at most countable) subset Bg. In the next move,player I again plays a maximal antichain A1 above p, and player II is now allowedto play two countable sets: BJ ~ Ao, B} ~ A1 •

In the n-th move, player I plays a maximal antichain An above p, and playerII plays countable sets Bli ~ Ao, . . , , B~ ~ An.

I II

Ao B8A1 BJ ,B}A2 B5,B?,B~

After W many moves, player II wins if there is a condition q ? p such that,letting En := U~n B~,

74 MARTIN GOLDSTERN

(See the end of section 1 for the definition of Afq.)

Definition 2.4. IP' is proper iff for all p E IP' player II has a winning strategy inthe game Gac(lP',p).

Note that by this definition, ccc forcing notions are trivially proper, since wecan play B:; := An.

Why are we so interested in antichains? Recall that one of our goals is not tocollapse WI . SO we have to deal with sequences (gn : nEw) of ordinals in theextension. Now it is well-known that the information in a name of an ordinal isreally coded in an antichain, as follows:

Definition 2.5 . Let A ~ P be a maximal antichain above pEP, and let f : A ~Ord be a function. Then g := gA ,! := {(J(q),q) : q E A} is a name for an ordinalabove p (i.e., p If- gA,! E Ord), and for each q E A we have q If- g = f(q) .

This definition shows us how to translate an antichain (plus an enumeratingfunction) into a name of an ordinal. Conversely we can translate a name of anordinal into a function defined on a maximal antichain:

Definition 2.6. Whenever p If-p 13 E Ord, we define D{3 (or more precisely, D{3,pas follows: - - -

DI} := {q ~ p: 3'Yq 'Yq If-~ = 'Y}

Clearly, D{3 will be dense open above p, and whenever A ~ D{3 is a maximalantichain in D{3 then the function f which maps each q E A to 'Yq carries allinteresting information about the name ~, in particular: p If-~ = gA,!'

Using this correspondence we can now translate the antichain game to thefollowing game:

Definition 2.7. Let IP' be a forcing notion, p E IP'. The (unrestricted) ordinal gameGor(IP', p, 00) is defined as follows:

In the n-th move, player I plays a IP'-name gn of an ordinal (above p, i.e.,p If- gn E Ord). Player II reponds with a countable set B n ~ Ord.

After w many moves, player II wins if there is a condition q ~ p such that,letting B := UkEW Bn we have

q If-p 'lin gk E B

For any ordinal X we also define the game Gor(lP',p,X) which is similar toGor(lP',p, 00), but player I has to play names gn for which p If- gn < X holds, andplayer II responds with countable sets Bn ~ X.

Remarks 2.8. First, note that the set B is countable, so if player II has a winningstrategy in Gor(lP',p, 00) then we have:

If p If- {1 ~ Ord is countable, then there is q ~ p and a countable set B suchthat q If- {1 ~ B .In particular, If-p NI v = NI v

P

, and more generally the property cf(o:) > No ispreserved when passing from V to V P•

Next, note that if player II has a winning strategy, then he also has a winning strategy in which all sets Bn are singletons (by dividing w into countablymany countable sets, and using a simple bookkeeping method) . Furthermore, allowing player I to play countably many ordinals in each move does not change theexistence of a winning strategy for player II, either.

Finally note that a winning strategy for player II in the game Gor(IP, P, X)for any large enough X (say, X > WI) will give a winning strategy for player IIin the antichain game, using the correspondence between names of ordinals andantichains discussed above. Conversely, a winning strategy for the antichain gamegives a winning strategy for any Gor(P, p, X) (including X =(0).

Also, note that if a forcing notion is e-closed, we can easily describe a winningstrategy for the game Gor(lP,p) : Player II will construct an increasing sequence(pn : nEw), p ~ Po ~ PI ~ . .. and a sequence (t3n : nEw), such thatPn II- gn = t3n . After W many moves, any upper bound q of the sequence Pn : nEwwill force "In gn = t3n .

We now give another characterisation of properness, which may at first lookrather complicated, but turns out to often be easiest to verify in actual applications:

Definition 2.9. Let X be a "large enough" regular cardinal. (It will turn out thatthe property we define will not really depend on X). We write H(X) for the familyof sets whose transitive closure has cardinality < X. (H(X) satisfies all of ZFCexcept possibly for the power set axiom.) Let (N, E) be a countable elementarysubmodel of (H(X) , E), and let lP' EN be a forcing notion.

We say that q E lP' is (N, P) -generic (or, when we are lazy: "N-generic" or"P-generic" or just "generic"), if the following two (equivalent) conditions hold

- Whenever A ~ P is a maximal antichain, A E N , then Afq ~ N . (See theend of section 1 for the definition of Afq.)

- Whenever g EN, and II-p g E Ord, then q II-p g E N (i.e.: whenever G ~ Pis generic over V, q E G, then g[G] EN), or in shorter notation:

q II- N[G] nOrd = N nOrd.

(The equivalence between those conditions can again easily be shown using thecorrespondence between names of ordinals and antichains that we discussed above.)

Since we have already defined properness, we will call the following propositiona "theorem". Alternatively, we could have used it as definition of properness.

Theorem 2.10. lP' is proper iff: for all (or some) large enough regular X, for allelementary countable submodels (N, E) -< (H(X), E) containing lP' and allp E lP'nNthere is a condition q E lP', q ~ p, such that q is (N, lP')-generic.

We leave the details of the proof to the reader, but we will give the followinghints. If you have a strategy 0" for player II in the game Gac(lP',p) then there willbe such a strategy in N. Let player I play all antichains A which are elements of N- after all, there are only countably many! Player II will respond with countable

76 MARTIN GOLDSTERN

subsets B n which are in N (since a as well as Ao, ... ,An are in N) Now sinceeach Bn is countable and an element of N, we must also have Bn ~ N .

Conversely, we can define a strategy by letting player II in the n-th step construct a countable model N n containing No, ... , Nn- I as well as Ao, . . . , An aselements, N n ~ H(X) . After W many steps, let N = UnEw N n, then any (P,N)generic condition will witness that player II has won.

We already know that a-closed forcing notions are proper, as are all forcingnotions satisfying the countable chain condition. Later we will see that propernessis preserved under composition of forcing notions, so also any finite compositionof forcing notions satisfying the ccc or a-completeness (e.g., Mathias forcing) isproper.

The following example is essentially different from those "trivial" examples,and is quite typical for a large class of proper forcing notions:Example 2.11. Let IP' be the forcing notion "adding a club to WI with finite conditions" . Conditions in IP' are finite strictly monotone partial functions p from WI toWI. It is easy to see that the sets {p : maxdom(p) > a} are dense for all a E WI, soa generic filter G will add a monotone function 9 from some unbounded B ~ WIto WI .

Let C be the closure of ran(g), then C is a "generic" club set.We claim that IP' is proper. So let (N, E) ~ (H(X), E), pEN. Let d:= N nWI .

Clearly d is an ordinal (recall that every countable element of N must be a subsetof N) . Note that dom(p) and ran(p) are finite sets in N, so they must be subsetsof N and hence of d.

Hence q := p U{(15, c5)} is a monotone function, so q E IP'. We claim that q isgeneric. So let A E N be a maximal antichain. We want to show that Afq ~ N,so towards a contradiction assume that there is rEA \ N, r compatible with q.

So for all a E dom(r), r(a) < d iff a < d. Let r' := rnN, then also r' E IP'nN.Since N F A is a maximal antichain, we can find a E AnN which is compatiblewith r' . Now check that a must also be compatible with r, since for all pairs (a,f3)in r \ r' we have a,f3 > 15. But a E AnN, rEA \ N and A is an antichain, so aand r cannot be compatible, a contradiction.

3. Variants of properness

It can happen that a forcing notion just "barely misses" being proper, but still hasmany of the good qualities enjoyed by proper forcing notions. In this section wewill first give an example of a nonproper forcing notion that is "almost" proper,and moreover "almost" a-complete, and then describe a variant of properness thatis satisfied by this forcing notion.

Definition 3.1. Let S ~ WI be stationary. Define

IP's := {J : 3a < WI dom(J) = a + 1, f increasing continuous, ran(J) ~ S}

with the natural ordering: f s 9 iff f ~ g.

lP's is called "collapsing Wi \ 8 (or: shooting a club through 8) with countableconditions". Indeed, it is easy to see that a generic filter on 8 will induce anincreasing continous map f from Wi into 8 , so in VPs the set Wi \8 will be disjointfrom the club set ranf, hence nonstationary.

We will see below that lP's cannot be proper (at least if Wi \ 8 is stationary inthe ground model) . On the other hand, lP's is almost a-complete, as the followingremark shows:Remark 3.2. Consider a sequence fo :$ It :$ h :$ ... of conditions, let dom(fn) =an + 1, and without loss of generality, assume an < an+! for all n. Let

a := sup{an : nEw}

It is now easy to see that

- If § E 8, then (fn : nEw) has a (least) upper bound, namely, Un fnU{(a, §)}- If § ~ 8, then (fn : nEw) has no upper bound.

Thus, in "many" cases we have a version of a-completeness.We use this example to motivate the following definitions:

Definition 3.3. Let 8 ~ Wi and suppose lP' is a forcing notion. We say that lP'is8 -proper iff:

For all (N, E) -< (H(X), E), if N is countable with N n Wi E 8, then for allp E lP' n N there is q ~ p which is N -generic.

Thus, we demand the existence of an (N, lP') generic condition not for allmodelsN but only for a certain (stationary) subset of the set of countable elemetarysubmodels of H (X).

Definition 3.4. 1. Let (N, E) -< (H(X), E) and suppose lP' E N is a forcingnotion. We say that q E P is N -complete if for all dense open sets DENthere is pEN n D, p $ q, or in other words, if the set

{pEN :p:$q}

is an N -generic filter on lP'.2. Let 8 ~ Wi and suppose lP' E N is a forcing notion. We say that P is 8

complete iff:For all (N, E) -< (H(X), E), if N is countable with NnWi E 8, then for allp E lP' n N there is q ~ p which is N -complete.

The following facts are immediate consequences of the definitions.Fact 3.5. 1. If q is N-complete, then q is N-generic.

2. If lP' is 8-complete then lP'is 8-proper.3. If lP' is a-complete, then lP'is 8-complete for every 8 .4. If lP' is proper, then lP' is 8-proper for any 8. (More generally, properness =

Wi-properness, and if 8 ~ 8' then 8'-properness implies 8-properness.)Note that N-completeness is much stronger than N-genericity : An N-complete

condition decides all names g of ordinals which are in N, whereas an N -genericcondition merely forces that they will be interpreted somewhere in N.

78 MARTIN GOLDSTERN

We leave as an exercise to show that if IP' has the countable chain condition,then every condition q E Pis N-generic (whereas typically N-complete conditionswill not exist in such cases).

Finally weshow that the notion of S-completeness is appropriate for the forcingnotion we have defined above:Fact 3.6. IP's is S-complete.

Proof. Let N --( H(X), 8 := Nnw!, p E IP'nN, and assume 8 E S. Let (Dn : nEw)list all dense open subsets of IP's which are in N. In particular, for every a < 8 thislist will contain the set

Ea := {p E IP': maxdom(p) > a} n {p E IP': maxran(p) > a}

Define an increasing sequence (fn : nEw) of conditions satisfying p $ Po andPn E Dn n N for all n. Let an := max dom f n' 8n := maxranfn. Clearly 8n < 8(since t« EN), and the sequence (8n : nEw) cannot be bounded below 8. By 3.2the sequence (fn : nEw) has an upper bound f . Check that f is N-complete. 0

We now show that S-properness is still sufficient to ensure that WI is notcollapsed. Moreover, weshow that all stationary subsets of S will remain stationaryin vP if IP' is S-proper.

(This will imply that IP's cannot be proper unless WI \ S was nonstationary: IfWI \ S was stationary, then IP's does not preserve its stationarity.)

Theorem 3.7. Assume that S is stationary, and IP' is S-proper. Then:

1. If-p WI vP =WI V

2. II-p S is stationary.3. Whenever a is an ordinal with uncountable cofinality, then If-p "a has un

countable cofinality".

Proof. We prove only (2). (1) and (3) are easier.So assume that p If-p "Qns = 0, Qa closed unbounded set" . We can find a name

f of a strictly increasing continuous function from WI to WI with P If-p ran(f) =Q.- Now we choose a model N --( H(X) satisfying NnWI E S, where N contains allnecessary information, such as IP', t, p, etc. [Why is there such a model? Rememberthat S is stationary. Start with acontinous tower (Ni : i < wt} of elementarysubmodels and use the fact that {N i n WI : i < wI} is a closed unbounded set.]

Let 8 := N nWI, so 8 E S. By S-properness we can find q ~ p, q N-generic.We claim that q If-p 8 = sup(Q n 8), which easily gives a contradiction.

If this were not the case, we could findr ~ q and a < 8 (so a E N) such thatr If- Q n8 ~ a . But as a E N, we can find a name {3 E N with If-p f(a) = {3. Sinceq is N-generic, q If-p {3 E N . So r If- a $ f(a) = {3 <: 8, {3 E Q, a contradiction.

This concludes the proof. - - -o

(More generally, it is easy to check that C E N implies N n WI E C wheneverN < H(X) and C ~ WI is club. It is also true (for any generic extension) that

(N, E) -< (H(X)v, E) implies (N[G], E) -< (H(X)vP

, E), if only X is sufficientlylarge compared with P. [Here, N[G] is defined to be {;p[G] : ;p EN} .]Now for any Y-generic filter G we have Q[G] E N[G], so N[G] n WI E Q[G] . If Gcontains in addition the N-generic condition q, then we also have N[G] n WI =N n WI =fJ E 8, which leads to a contradiction with II-p Q n 8 =0)

Why are we so obsessed with this complicated property "properness" and itsvariations, when we are really mainly interested in the seemingly simpler propertiesof "not collapsing WI" or "preserving certain stationary subsets of WI "?

The reason is that in many applications we need to construct a forcing notionthrough (finite) composition and (transfinite) iteration of various simpler forcingnotions. However, the property "not collapsing WI" may not be preserved in limitsteps of such iterations. That is, if we define a sequence

Po = {0}PI = Po *Qo, where Qb is a Po-name of a forcing notionP2 = PI * (~h, where ~ is a PI-name of a forcing notion

Pn+I = Pn*9n

then it is possible that all the forcing notions Pn (in Y) and all the forcing notionsQn (in the respective ylPn) preserve Nl , but there is no conceivable "limit" Pw

~hich will also preserve Nl .

Example 3.8. Partition WI into W many disjoint stationary sets WI = 80 U81 u· ...Let Qn be the forcing notion PWI \Sn in y Pn, where Pn is defined as above. Thenif y' 2 y is any universe in which there are Y-generic filters for all Pn, then

y ' F= WI V is a countable union of nonstationary sets, hence nonstationary

so in y' WI is countable.

The raison d'etre for properness is the following theorem:

Theorem 3.9. Properness is preserved in countable support iteration.That is, assume that

1. (Pa ,Qa : a < fJ) is a countable support iteration with CS limit Po. (I.e .,each Pais the set oj all partial junctions with countable domain ~ a andpfa II-Pa p(a) E Qa).

2. For all a < s, Il-lP a 9a is proper.

Then jor all a < fJ, Pais proper.

Shelah's original proof of this theorem can be found in [9, chapter III]. Alternative proofs are in [4] (repeated in [3]) , or (using games) in [9, chapter XII] and[5].

Similar proofs show the analogous thereom with "proper" replaced by 8-proper,for any 8 ~ WI.

80

4. Semiproper iteration

MARTIN GOLDSTERN

Definition 4.1. Let r be a forcing notion, X a sufficiently large regular cardinal,(N, E) -< (H(X), E), PEN. We say that q E r is (N, r)-semigeneric, if q I~p

N[G]nWlv =Nrua, v, i.e., for all names g in N, ifl~p g E Wl v, then q I~p gE N .

Definition 4.2. We say that r is semiproper if and only if for all r and N asabove , for all pEr n N, there is q ~ p, q (N, r)-semigeneric.

Equivalently, r is semiproper if player II has a winning strategy in Gor(r,p, Nl )

for all pEr. (The equivalence can be shown as in 2.10.)

Why are we interested in semiproperness? Semiproperness is a weak version ofproperness which is still sufficient to show that Nl is not collapsed (the same proofworks). Moreover, semiproperness is preserved in some iterations. The differencebetween properness and semiproperness is that 3.7(3) is in general (for a > Wl)not true for semiproper forcing, see 4.4(2).

There can be useful forcing notions which are semiproper but not proper, asthe following example shows.

Example 4.3. Let K. be a measurable cardinal, D a normal ultrafilter on K. . Prikryforcing r D is defined as

rD := {(s ,A) : s E [K.]<W, A E D,maxs < min A}

We let (s, A) ~ (t, B) (remember the alphabet convention) iff s ~ t, B ~ A, andt \ s ~ A (so t is an end extension of 8).

The following fact is well known:Fact 4.4. 1. For all A < K., r D does not add new subsets of A.

2. I~PD "K. is a cardinal of cofinality No".This implies that rD does not collapse Nl but is not proper. We will show that

r D is semiproper.

Lemma 4.5. Let D be a normal ultrafilter on K. . Let g be a r D-name for anordinal, (so,A) E rD, (so,A) I~PD g < Nl .

Then there is a countable set B and a set A' ~ A, A' ED such that (so,A') I~PD

gEB.

Proof. For notational simplicity we will assume that 80 = 0.For each 8 E [K.]<w let As, f3s be such that

- if there is A* ~ A, f3* such that (s, A*) I~ g = f3*, then (As, f3s) is such a pair;- otherwise f3s = *.

For a < K. let ACt := nsE[Ct+lj<w As, and let All: be the diagonal intersection of allthe ACt: All: ={i E K. : (\la<i)(i E ACt)}.

Then All: E D (because D is normal), and for all a < K., for all 8 E [a+ l]<Wwe have All: \ (a + 1) ~ ACt ~ As.

By Rowbottom's theorem we have the partition relation K. -+ (K.)~~ andmoreover we can find the homogeneous set in D . (That is, given any function

f : [I\:]<w -t WI there is A' E D such that for all nEw, Jr[A']n is constant.)So there is A' ~ A", A' ED, and for all nEw there is an E WI such that

Vs E [A']n as = an

Let B := {an: nEw}. It is now easy to check that (so, A') I~ g E B. 0

Using this lemma, we can now win the semiproperness game by playing anincreasing sequence of pure extensions Pn = (so, An) bounding more and morenames of countable ordinals.

Unfortunately semiproperness is not preserved by countable support iteration.This has a very natural reason, which can informally be explained as follows:Consider a countable support iteration (IP'0, Qo : a < 1\:) with CS limit IP'". Assumethat cf(l\:) > No, but that this property is "lost" during the iteration, say IP'I I~cf(l\:) = No . (E.g ., if Qb is Prikry forcing.) Now IP'" is the direct limit of (IP'0 : a < 1\:)but if we want to construct a semigeneric condition for IP'" we would like to usea condition with unbounded support, i.e., we would like to put information on anunbounded (hence uncountable) set of coordinates.

Revised countable support (RCS) iteration circumvents this problem by (stillinformally speaking) allowing unbounded sets as supports, as long as they are atleast countable in some intermediate universe . In other words, we allow the supportof a condition to be not only a countable set, but even a name of a countable set.

(There is no need for such tricks when we deal with proper forcing, becausecountable sets in any proper forcing extension are covered by a countable set fromthe ground model.)

Definition 4.6. Let r = (IP'0, Qo : a < 8) be an iteration, 8 a limit ordinal. TheRCS-limit of r is defined as the-set of all p E proj lim r (= the projective or "full"limit of r) satisfying

Vq E proj lim Jii>, q 2:: p implies that there is a < fJ and r 2:: qfa withr 1~"cf(8) = No or supp(p) n (a,8) = 0."

Here, supp(p) n (a ,8) is a 1P'0-name for the set {,B E (a,8): pf(a,8) .wIPQ , ~ p(,B) =0Q

Q, and IP'0,0 is the "quotient forcing 1P'0 : IP'0" (which can be shown to also be the

result of an RCS iteration.

Definition 4.7. We call an iteration Jii' = (IP'0, Qo : a < 8) an RCS-iteration, iffor all a < 8 we have IP'0 = RCS lim(Jii'fa) . -

(For technical reasons we also define RCS lim IP' if 8 is a successor ordinal,8 = ,B + 1. In that case the RCS limit is essentially equal to 1P'13 *Q13 - formallywe define it to be the set of all functions p with dom(p) ~ (3 + 1 and either p E 1P'.aor pf,B EIP.a and pf,B I~P /I p(,B) E 9 13'

This definition of RCS iteration is from [8]. An equivalent definition (in thelanguage of Boolean algebras) can be found in [1] , [2] . The original definition is in[9, chapter X].

Theorem 4.8. RCS iteration preserves semiproperness.

Proof. See the papers quoted above . o

82

References

MARTIN GOLDSTERN

1. Dieter Donder and Ulrich Fuchs, Revised Countable Support Iteration, in: Handbook of Set Theory, (Matt Foreman, Akihiro Kanamori, Menachem Magidor, eds.)Kluwer, to appear.

2. Ulrich Fuchs, Donder's version of Revised Countable Support, PhD thesis, FreieUniversitat, Berlin (See also ftp://ftp.math.ufl.edu/pub/logic/fux/rcs.tex.)

3. Martin Goldstern, Tools for Your Forcing Construction, in: Set Theory of The Reals(Haim Judah, ed.), Israel Mathematical Conference Proceedings 6 (1993) 305-360.

4. Martin Goldstern and Haim Judah, Iteration of Souslin Forcing, Projective Measurability and the Borel Conjecture, Israel Journal of Mathematics 78 (1992) 335-362.

5. C. Gray, Iterated Forcing from the Strategic Point of View, PhD thesis, Universityof California, Berkeley, 1982.

6. T. Jech, Set theory, Academic Press, New York, 1978.7. K. Kunen, Set Theory: An Introduction to Independence Proofs, North Holland,

Amsterdam, 1980.8. Chaz Schlindwein, Simplified RCS iterations, Arch. Math. Logic 32 (1993) 341-349.9. Saharon Shelah. Proper Forcing, Lecture Notes in Mathematics 940, Springer

Verlag, Berlin-New York, 1982.10. Saharon Shelah, Proper and Improper Forcing, Perspectives in Mathematical Logic,

Springer-Verlag, 1998?

APPLICATIONS OF p-FUNCTIONS

PIOTR KOSZMIDERlDepartment of MathematicsAubum UniversityAuburn, AL 36849-5319USA

The exponential function. This is undoubtedlythe most important function in mathematics.

Walter Rudin; Prologue to Real and Complex Analysis

o. Why do we need p-functions?

Once we start measuring mathematical objects using infinite cardinals we are lednaturally into two-cardinal combinatorics which is a field about combinatorial constructions with associated two cardinals. Various internal and external questionscan be asked, all related to the relation between the two associated cardinals, e.g.:

What could be heights of superatomic Boolean algebras with countable width?

What are the possible sizes of Hausdorff spaces with points Go and countableLindelof degree?

Is it possible to construct a c.c.c forcing notion (and in particular a cardinalpreserving forcing notion) that adds a function f : W2 x W2 ~ W which isn'tconstant on a product of any two infinite sets?

If we are given a construction with property P of size K, and we are still ableto perform it to get a construction with property P of size K,+, we say that westepped-up some property of K, to K,+, namely the property of the existence of anobject of size K, with property P.

Let us consider the above questions. It is trivial to get a superatomic Booleanalgebra of countable height and countable width. Juhasz and Weiss ([JW]) steppedup this construction to Wl, i.e., they obtained a superatomic Boolean algebra of

IThe author was partially supported by the NSF Grant DMS-9505098. The authorwas also partially supported during the conference by Centre de Recerca Matematica(CRM) for which he is very grateful to the organizers of the conference: Joan Bagariaand Adrian Mathias.

83


84 PIOTR KOSZMIDER

countable width and height WI (actually any a < W2) ' Baumgartner and Shelah([1]) stepped-up this construction to W2 and it remains open whether it can bestepped up to W3.

Regarding the second question, it is clear that the reals give an example ofsuch a space of size 2W

• It was Shelah ([16]) who stepped-up the construction to(2W )+. Later 1. Gorelic ([4]) stepped it up to 2W1

, regardless of the size of 2w , andrecently C. Morgan ([14]) extended this result to n-th beth. The last constructionwas stepped-up by S. Todorcevic ([19]) to W2 and it remains open whether it canbe stepped-up to higher cardinals.

Stepping-up is the heart of two-cardinal combinatorics. Various tools can be usedto step-up properties of cardinals; here are a few examples:

HAUSDORFF GAP: was used by Hausdorff among others to prove the existence ofan uncountable universally measure zero set. Thus a property of countable sets isstepped up to WI

CH : used e.g., by Sierpinski and Luzin to get other special sets of the reals bystepping-up properties of the countable .

KUREPA TREES: used by Kurepa e.g., to step-up properties of linear orders .

MORASSES: used originally by Jensen to step-up properties of the countable inmodel theory.

0 : used by Jensen to get a nonreflecting stationary subset of W2 stepping up theproperty of WI itself.

A FUNCTION WITH THE ~-PROPERTY: used originally by Baumgartner and Shelahto get a superatomic Boolean algebra of height W2 and width w.

There are also many new results using the above stepping up tools:

(M.Rabus [15]) using a function with ~-property got a countably tight noncompactinitially WI-compact space of size W2.

(P. Koszmider [9]) If there is a Kurepa tree and CH holds then there is a regularnonmetrizable space all whose subspaces of size WI are metrizable.

(P. Koepke & J .C. Martinez [12]) If there is a (11:, l l-morass then there is a superatomic Boolean algebra of height 11:+ and width 11:.

p-functions can be considered as another stepping-up tool for stepping from II: to11:+. Their strongest properties are exhibited under the assumption of only OK'In fact the underlying idea of p-functions is S. Todorcevic's Method of MinimalWalks. p-functions are tools for performing this method on square sequences.

The advantages of p-functions are as follows:

APPLICATIONS OF p-FUNCTIONS 85

1. p-functions provide canonical methods for stepping up. Sometimes to get thefull strength of the result one needs to assume that the underlying sequenceis a square sequence.

2. Most of the above stepping-up tools can be canonically reconstructed in somesense from p-functions.

We will provide below examples of these canonical methods and by this we wouldlike to defend the statement that

p-functions are the most important functions of two-cardinal combinatorics.

1. What is a p-function?

The exponential function is given by the following formula eZ = E~=o ~~

Walter Rudin; Prologue to Real and Complex Analysis

For a regular cardinal «, the definition of the p-functions assumes that we aregiven a sequence (Co: a < ",+) with the following properties: Co is closed andunbounded in a, Co+! = {a} and an ordinal that occupies a successor place inCo is not a limit ordinal. Given this sequence we can consider minimal walks fromone ordinal {3 E ",+ to another ordinal a < {3. The walk is given by

{31 =min(C,B - a) , {32 =min(C,BI - a), ...

until we reach a which must happen after finitely may steps as the above walk is afinite decreasing sequence of ordinals not less than Q . Note that we reach a whenand only when a E C,Bn .

~a. ~1 ~'-----7

,-- ) :tK

Cp

~I

86 PIOTR KOSZMIDER

~a. ~2

f-----+---+--~) :to1C

CP2

We most often assume that the sequence (Co: : a < K,) is a square sequence i.e., werequire also that tp(Co:) $ K, and if {3 is a limit point of Co: we have C{3 =Co: n{3.So for example if (3 E Lim(Co:), then the walks from {3 to 'Y < {3 and from a to 'Yare the same.

p-functions are functions that are supposed to capture some essence of the minimalwalks, so any p of a, {3 is some object associated to the walk from (3 to a. E.g.

po(a, (3) = (tp(C{3 n a)~po(a,min( C{3 - a)))

po(a,a) = 0

So Po (a, (3) is made of order types of parts C{3i - a where {3i's are from the minimalwalk from {3 to a. It makes sense in ZFC, even if (Co: : a < K,+) is not a squaresequence, note that

Stevo Todorcevic has defined several p-functions, all coding some properties ofminimal walks. We will focus our attention on one of them called just p. We defineit e.g. by

p(a, (3) =sup] tp(C{3 n a), p(a, min(C{3 - a)), p(~, a) : ~ E C{3 n a}

pio, a) = 0

In a vague language p(a, (3) is the supremum of several ordinals involved in thewalk from {3 to a. Note that if we assume that (Co: : a < K,+) is a square sequencewe have that tp(C{3 - a) < K, and so

and directly by the properties of minimal walks we have

for'Y < a < {3 such that a E C~ .


Because of limited time we will not discuss a beautiful dynamics of these kind ofdefinitions that lead to many other important functions. For these discussions see[18], [2].

2. Properties leading to the canonical methods

... any mathematical argument, however complicated, must appear to me as a uniquething. I do not feel that I have understood it as long as I do not succeed in graspingit in one global idea.. .

J. Hadamard; The Psychology of Invention in the Mathematical Field

What is stepping up all about? From some point of view it is about getting aconstruction of size 1\;+ in x-many steps, meaning that during the constructionwe face only less than I\; previous stages. So how to get to a high window havingonly short ladders? Connect the short ladders together, and move from one to theother. This idea is behind morasses as well as square sequences: homogeneity ofmorasses or coherence of square sequences is the way we connect the ladders withinitial fragments of length less than 1\;.

When dealing with a coloring p : [1\;+]2 ~ I\; as a stepping-up tool we should beseeing representation of each ordinal e E 1\;+ as I\; namely

l.e., we are considering the level sets of pC ,e) : e~ 1\;. Some versions of p-functionsare even one-to-one in that sense that the above level-sets are at most singletons.

A fundamental property that is leading to some canonical methods is expressed inthe following fact.

88 PIOTR KOSZMIDER

Fact. There is r : [W2]3 -t W2 such that 0 < f3 ~ r(o, f3, '1) ~ '1 such that wheneverX E [W2]Wl is of order type Wl and e =sup X, then there is X' E [X]Wl such that

'I/o < f3 < '1, 0, f3,'1 E X' Pc. r(o, f3, '1)) IX' n'Y = p("c)1'Y n X'

Before the proof, let's note the meaning of this fact . It says that given a subset ofW2 of order type Wl, we can associate some points of view r(o, f3, '1) to every tripleof this subset . After thinning the set out we get that all these points of view arecoherent. Here the point of view 0 < W2 means the enumeration pC, 0) of o.

Proof. Define r(o,f3, 'Y) to be the last place where the walk from '1 to 0 agree withthe walk from '1 to f3.Consider Ceo Clearly it is of order type Wl. Thin out X to Xl and find a clubof limits D ~ O; so that the elements of D and Xl come alternately. Call ~.s anelement of Xl above 0 E D such that for all 0' E D 0' > 0 we have e.s < 0', so thatthey look like:

IS I I II ,

Now pick up an c.s such that it is the last element in the minimal walk from e.s tooabove o. Note that then we have 0 E Ce& 1 and as 0 is a limit we have that aisa limit in Ce& as well. Moreover for all previous elements ~ in the minimal walkfrom {.s to 0 did not have this property i.e., C( no were bounded in a.

Then by the coherence of the OWl -sequence we have that

Now we can press down the bounds of C( na for fs in the walks from {.s's to a'sfor which these sets are bounded Oust fs in these walks above c.s's). This pressingdown results in a stationary S ~ D 1 such that for any 0, '1 E S the walk from e')'to ~Q goes through c')',

~I I . , I

~ l;a ~ Sfj

and so

because both of the walks from ~1' to ~a and from ~1' to ~/3 go through c1' and theysplit after c1' (by the fact that (3 E Ce., which follows from the coherence of thesquare sequence and because elements of Xl are separated by elements of D) forevery a,(3,'Y E S, but p(',c1')I'Y = p(',c)I'Y, as required. 0

Note that in the above proof we very much utilized the fact that a simple definitionof p is at hand and an extra property can be proved when needed. That is notso often the case with coloring of squares obtained from Velleman's morasses [21](first investigated by Charles Morgan [13]), for example see [22]. Sometimes inthis case one even needs to force the coloring to use the genericity ([1], [15]) or useheavily elementary submodels and CH [10].

Now let us bring up other fundamental properties of p.

Fact. Let a < (3 < 'Y < ,,;+, v < n, 0 < 0 = uo < e < ,,;+, then the followingconditions are satisfied.

(a)

(b)

(c)

p(a,'Y) ~ max{p(a,(3),p«(3,'Y)}

pea,(3) ~ max {pea,'Y), p«(3, 'Y)}

(d) there is a « 8 such that p(~,c) ~ p(~,8) for all (~~ < 0

Let us give a reason why one of the above holds, e.g., b), the reason is that thewalk from 'Y to (3 and then from (3 to a is a walk from 'Y to a, so the minimalwalk from 'Y to a should give a smaller value of pea, 'Y) then max{p(a, (3), p«(3,'Y)}.For the proofs see [18] or [2]. Let's build some intuition about these propertieslooking at our picture of p. The first property says that the level sets are small,in particular for p : [W2J2 -t WI we have pC,a) : a -t WI is countable-to-one . Thesecond two properties express in what sense p can be used for measuring distanceof ordinals of ,,;+ using ordinals < ,,;. Here we exactly see how the properties of themaximum function on [,,;]2 are stepped-up. The last property partially capturesthe preparatory work for pressing down that we witnessed in the previous proofand is usually used together with the pressing down lemma.

3. The first canonical use: direct stepping up

What is p? It is not just a function with properties (a)-(d)j it is an approach tostudying the walks along square sequences

From M. Bekkali; Topics in Set Theory;Notes on lectures by Stevo Todorcevic.

Let us consider the following notions considered by Erdos and Hajnal: Fix a function f : [,,;]n -t ,,; such that f(a) 't a. A set X ~ ,,; is called free for f if and only

90 PIOTR KOSZMIDER

if f(a) ft X for any a E [XJn. One can naturally ask whether for every functionas above there is an uncountable free set. Much effort has been put in obtainingcounterexamples to the above assertion. In L they have been obtained for n =3,also sophisticated forcing notions as wellas morasses were used to construct them.Here we present a simple construction due to Todorcevic. (The same proof givemuch stronger conclusion that solved other open problems, we refer an interestedreader to [18J and [2J.)Fact. If onholds then there is f : [wd 2 ~ WI such that for every uncountableX ~ WI, j"(X) =WI'

Actually, another fine analysis of minimal walks yields one of the most strikingresults in set theory which says that the assumption of CH can be dropped . Herewe assume CH because it is very easy to prove the fact under this assumption andanyway we are interested in using it in L .It follows that there is a function f : WI ~ WI in L for which there is no uncountablefree set. Now we do stepping-up using OWn .

Fact. Suppose n ~ 2. If there is a function f : [II':Jn ~ II': for which there is nouncountable free set, then there is a function f : [1I':+]nH ~ 11':+ for which there isno uncountable free set

Proof. Let c : [II':]n ~ II': be as in the assumptions. For an n + I-tuple a of 11':+ definea coloring d in the following way: Transport the n + 1 tuple a into 11':, for this user(o:, {3, "y) where 0:, {3, "y are the last three elements of a, i.e., consider

A = p(",r(o:,{3,"Y»"(a - h})

Now apply c to A and obtain an element c(A) of 11': . And now find d(a) in "y suchthat

p(c'(a), "y) = c(A)

If there is no such element put c'(a) =O.

K

Now let's prove the fact. Take an uncountable X ~ 11':+, we may w.l.o.g. assumeby the previous results that X is of order type WI and that for every 0: < {3 <

'Y < 1\;+ we have p(',r(a,(3,'Y) ~ pCe), where e = sup(X) . That proves thatour transportation of n + l-tuples from X' was done always according to p(., e).So consider p(X') ~ I\; which is uncountable. Now by the assumption there isA E [p(',e)(X')]n such that c(A) E X', so now it is easy to find a E X' such thatd(a) E X' as required. Here we cheated a little bit: we assumed that pC e)(X') is1-1, which is not the case. But it is easy to modify it, preserving equalities of thesort p(a,'Y) =p(a,(3) this is done in a canonical way by putting

p*(a,(3) = 2P(a ,.B) · (2· tp{~ ~ a : p(~,a) ~ p(a,(3)} + 1)

where the operations are ordinal multiplication and exponentiation.o

So this result of Todorcevic proves that in L there are functions f : [wn]n+l -+Wnfor which there are no uncountable free sets.

4. The second canonical use: stepping up in c.c,c, extensions

Sometimes p-functions can be used for constructing c.c.c. forcing notions which addconstructions that step up some properties to W2. Classical example of such a useof stepping up devices is a result of Jensen saying that OWl allows us to constructa c.c.c. forcing notion which adds a Kurepa tree (Generic Kurepa Hypothesis (see[5], [6])). On the other hand it is known that OWl does not imply the existence ofKurepa tree.p-functions can be used for this generic stepping-up as well. As far as I know,it was B. Velickovic who first used p-function in this context in his proof of theabove result of Jensen. The simplest property of p-function that we encounterwhile generically stepping up is the following:

Definition 4.1. A function f : [W2]2 -+ WI is called unbounded if and only if forevery family {a~ : ~ < wtl C [W2]<w of pairwise disjoint sets, for every a < WI

there are 6 < 6 < WI such that for every (31 E a~l and every {32 E a~2 we have

Fact 4.2. P : [W2]2 -+WI is an unbounded junction.

Proof. Suppose that {ae : ~ < wtl C [W2]<w is a collection of pairwise disjointsets. We may with out loss of generality assume that all the sets ae have exactlyn-elements for some nEw. Let us enumerate each ae as {ad1), ...,ae(n)} inthe increasing order. Suppose this collection does not satisfy the definition of anunbounded function for some a < WI, i.e., for every ~ < ~' < WI there are i,j E [n]such that p(ae(i), ae' (j)) < a. Fix U a uniform ultrafilter on WI . Using the standardargument find io,jo E [n] and A E U such that for every ~ E A we have

{~' : p(adio) ,ae'(jo)) < a} E U

Now consider two cases; first when there is ~ E A such that for U-many ~"s wehave ae'(jo) < ae(io). In this case we get contradiction with the property a) of the

92 PIOTR KOSZMIDER

function p. In the opposite case for every {,e E A there is U-many {"'s above ,{and e such that p(axdio), a{1I (jo)), p(adio), ae' (jo)) < 0 thus by the property c)of the function p we conclude that p(a{ (io), a{, (io)) < 0 for all {, {' E A but thiscontradicts another basic property of p which says that p hits WI colors on thesquare of any uncountable set (repeat the proof from section 2). 0

Let us present a construction due to Todorcevic which he used to prove that underM AW 2 , Chang's conjecture is equivalent to the statement which asserts that every9 : W2 x W2 ~ W is constant on a product of two infinite sets .

Fact 4.3. Assume OWl' There is a c.c.c. forcing not ion P which adds a 9 : W2 xW2 ~ W which is not constant on a product of any two infinite sets.

Before beginning the proof, one first needs to note the following claim.

Claim 1. Suppose f : [W2J2 ~ W is such that for every 0, {3 < W2 we have

I{{ < 0,{3 : f({,o) = f({,{3)}1 < w,

then there is 9 : W2 x W2 ~ W which is nonconstant on the product of any twoinfinite sets.

Proof of Claim 1. Define 9 as follows

g(a,p) = {

2

2/ (0 ,,8)+1 ,3,5/ (0 ,,8)+1 ,

5

if 0> {3;if 0 = {3;if 0<{3 .

Now, take two infinite sets A, B ~ W2 and consider three cases. The first when allelements of Ax B intersects two of the three parts of the domain of 9 exhibitedin the definition by cases. In that case, by the definition, the function is nonconstant. The remaining two cases are when A x B is included in the first orin the third part of the domain of g (note that it is never just the second part{(o,{3} E W2 x W2 : 0 = {3}). In these cases the property of g follows from theproperty of f as we have that either sup B < min A or sup A < min B.

o

It is not known (see [19]) whether 9 as above can exist on () x () for () > W2 , butone can prove that there is no f : [(}]2 ~ was above .

Proof of Fact 4.3. To prove the fact, we will show how to add f as in Claim 1.The forcing P consists of conditions of the form P = (aJIl fp) such that

i) ap E [W2]<w

ii) fp : [ap]2~Wiii) for all a < /3 < W2 we have

{~ : fp(~, a) = fp(~, /3)} ~ {~ < a, /3 : p(~ , /3) ~ p(a, /3)}

The order is defined as follows; P ~ q if and only if P 2 q and

iv) V~ E ap - aq Va, /3 E aq ~ < a, /3 => fp(~, a) =I f(~, /3)

Clearly the forcing adds a function as required. So let us check that P satisfies thec.c.c. Suppose (Pe : ~ < wd is an antichain. Put ape = ae. Thin out (Pe : ~ < wdso that

a) (ae : ~ < wd forms a ~-system with root ~.b) all fpe's agree on ~c) For all TJ < WI, for all a,/3 E~, a < /3 we have

(a11 -~) n {~< a,/3jp(~,(3) ~ p(a,/3)} =0

d) Fe's form a ~-system with root F, where Fe = {p(a',a)ja,a' E ad.

Now use the unboundedness of p to get P =Pel and q =Pe2 such that

e) '1/31 E ap - ~ '1/32 E aq - ~ p(/31,f32) > maxF.

To continue the proof, another claim is needed .

Claim 2. If ~ E ~ and a E ap - aq and (3 E aq - ap and ~ < a, (3, then

~ E {~< a, (3 : p(~,min(a,/3)) ~ p(a,/3)}

Proof of Claim 2. Suppose ~ < a < /3 (the other case is similar) . Suppose theopposite, i.e., that

(*): p(~, /3) > p(a, /3)

By (*) and p(~,a) ~ max(p(~,/3),p(a,/3))we get

p(~, a) s p(~, (3)

by (*) and p(~,/3) ~ max(p(~,a),p(a,/3))we get

p(~, /3) ~ p(~, a)

so p(~, a) = p({, /3) and so this ordinal is in F, but then bye) we have p({, /3) =p({,a) < p(a, /3) which contradicts (*). 0

94 PIOTR KOSZMIDER

Now, returning to the proof of the fact, let us see that p and q are compatible.Put ar =ap U aq, fr = fp U fq U f where f is 1-1 and ran(Jp U fq) n ran(J) =0.We check that r ~ p, q, rEP. We check iii) and iv) for all ~ < a < {3, ~,a, {3 Ear'

Case 1. a, (3 E ap '

If ~ ¢ ap , then iii), iv) hold by construction of f .If ~ E ap, iii) holds because pEP. iv) is not void only if ~ E ap- aq; a, (3 E apnaq.Then apply c) to get iv) by iii).

Case 2. a,{3 E aq •

Similar.

Case 3. a E ap - aq and (3 E aq - ap'Note that iv) is void. By the construction, iii) is not void only if ~ E .6.. By theclaim 2, p(~,min(a,{3)) < p(a,{3) and this trivializes iii) so iii) is satisfied.This completes the proof of the fact . 0

In fact the last case of the above proof as well as claim 2 contain the essence ofthe use of the unbounded function. If the conditions are defined using pairs buttriples are also involved, then when ~ is in the root of the .6. system and a, {3are in different tails, then the construction of the amalgamation cannot changeanything about pairs ~, a, ~,{3, so the triple ~ , a, {3 may not be appropriate forthe amalgamation. Now p-function captures such a ~ in a countable set. A clauseallowing the appearance of bad triples ~, a, {3, for these countably many ~'s maybe added to the definition of a condition of the forcing.

>

Let us examine this idea closer in the following general scenario for a possible useof the property of an unbounded function for the proof of the c.c.c. of some forcingnotion. (it is a modified version of a part of [8]) Suppose that we want to force,with finite conditions, a structure M whose underlying set is WI X W2, and whichis determined by its behavior on the elements of WI x W2 . Moreover we want theobtained structure to have the following property:

That is, for each pair {31, {32 E W2 the set of a's in WI which are bad for ({3I, (32) isfinite . Since we want the forcing to satisfy the c.c.c. there is a function m' : [W2]2 -+WI in the ground model which bounds bad a's. So, consider an unbounded function

as such a function, that is consider the following forcing notion, P consisting ofconditions P = (ap, bp,M p) such that

ap E [WIj<w and bp E [O]<w andM p is a finite structure of the considered sort, whose underlying set

dom(M p) is ap x bp.

If {3I, (32 E bp and nEap and n 2 P({3I, (32), then

M p F ({3,f32 ,n) is good

We write P $ q and mean that P is stronger than q if and only if:bp 2 bq, ap 2 aq and M q is a substructure of M p andV{3I,f32 E bqVn E (ap - aq) M p F ({3I,{32,n) is good.

Of course, the context in which we are working should allow us for proving thatP is a partial order and that for n E WI, {3 E W2 , the sets

are dense in P. Nevertheless the main problem could be the proof of the c.c.c, Solet us show how the fact that p is unbounded allows us to prove that P satisfiesthe c.c.c. Suppose the opposite.

Let A = {pe : ~ < WI} be an uncountable antichain in P. By the ~-system

lemma, we can w.l.o.g. assume that there are a E [wd<w and a pairwise disjointfamily {ae : ~ < wd ~ [Wd<WI such that max(a) < min(ae) for all ~ < WI andthat there are b E [W2j<w and pairwise disjoint {be : ~ E wd ~ [W2]<w such thatPe = (a U ae,b U be,Me) and Mela x b =Me'la x b for every ~,~' E WI .

Define no =max(a) . We may w.l.o.g. assume that p({3I,f32) < minae for everyae in WI and every {3I , {32 E b. Now using the fact that p is an unbounded functionwe can find 6, 6 < WI such that

{p((31, f32) : {31 E bel' f32 E b6} n no = 0

Call Pel' Pe2 as P and q respectively. We claim that the conditions P and q arecompatible. Since dom(M p) n dom(M q) = a x b, we should be able to concludein our context that M p and M q could be amalgamated to aMp U M q whoseunderlying set is ap x bpUaq x bq. Now we have to extend M pUM q to M definedon (aUael Uae2) x (bUbel Ube2) so that r = (aUael Uae2,bUbel Ube2,M) E Pand r $ p,q.

Define M on the remaining parts ael x be2 and ae2 x bel ' To construct M sothat the obtained condition is stronger that p we have to consider o E aq - ap and{31l {32 E bp. If ({3I , (32) ¢ [bq]2 then we have one free variable in the equation wewant to satisfy that is "M F ({3I,f32,n) is good" i.e., either ({3I,n) or ({32,a) isnot in the underlying set of M p UM q • So we should be able to solve the equationin our context i.e ., find an appropriate extension of M p U M q • If {31l {32 E bq , then{3I, (32 E b, and then we have no free variables, but then n 2 P({3I, (32) by the choiceof 6, 6, hence already M q F ({31, f32, o) is good . The idea of the proof that wecan extend the condition to a condition stronger than q is similar. Now we have to

96 PIOTR KOSZMIDER

show that the extension could be arranged in such a way that *) will hold. Again,we will exploit the possibility of the extension of MpUMq to a condition in which(/31, fh, a) is good whenever either (/31 ,a) or (fh, a) is not in ap x bp U aq x bq • So,note that either we have what we call above a free variable or already a given tripleis good in M p U M q• The only problem is when a E a, /31 E bp - bq, /32 E bq - bpor a E a, /31 E bq - bp fh E bp - bq , in those cases we have no free variables i.e,both (/31, a) and (/32, a) are in ap x bp U aq x bq , yet the triple belongs to neithercondition. But here we use the fact that p(/31' fh) > max(a) thus the condition *)does not address this triple, so we do not have to worry about it . Of course onlysolving all the above equations simultaneously can guarantee the existence of theappropriate amalgamation. This completes the description of the scenario.

Let us mention two recent uses of p for generic stepping-up.

Theorem 4.4. (See Eda, Gruenhage, Koszmider, Tamano, Todorcevic [9J) Using p and assuming oW2 one can define a canonical forcing notion that separatesclosed disjoint sets in the square of sequential fans times WI, F(w2)2 x WI. Thusassuming M.Aw3 we have the consistency of the normality of such a space (whichis independent).

Theorem 4.5. (See Koszmider [11].) Using p and assuming oW2 one can definea canonical forcing notion that adds a strong chain in p(wI)jFin i.e., a sequence(Xa : a < W2) ~ p(wI) such that Xa - X(3 is finite and X(3 - Xa is uncountablefor a < /3. For every two a < /3 < W2 we have that Xa - X(3 is included in p(a, /3).

5. Everything is p

In this section we list some results concerning the possibility of obtaining otherstepping-up tools from the p-functions. The p-function constructions tend to bemuch simpler than ones obtained without the use of it .

Theorem 5.1. (See Todorcevic [18J.) If 0(0) holds (for theta = /\:+ it is a weakversion of~), then the tree of restrictions of P2 (., a) for a < /\:+ with the inclusionis a O-Aronszajn tree.

It was previously proved by Jensen that~ implies the existence of a special x'tAronszajn tree. This result can also be canonically obtained using the p-functionand the above tree.

Theorem 5.2. (See Velickovic [20].) Using p and assuming OW2 one can define acanonical forcing notion that adds a Kurepa tree with finite conditions. Every twobranches with numbers a, /3 E W2 differ on the levels with numbers above p(a , /3) .

The above result was previously proved without p by Jensen.

Theorem 5.3. (See Todorcevic [18].) If D< holds and 2<" = «, then the tree ofrestrictions of pC ,a) for a < ",+ with the inclusion is a ",+ -Aronszajn tree.

Theorem 5.4. (See Todorcevic [18}.) Using p and assuming D< and 2<" = ", onecan canonically define a Pi-name for a ",+ -Suslin tree where P" is the standardforcing for adding a Cohen subset of «. In particular one always adds a Suslin treeby adding one Cohen real (a result obtained by Shelah using other methods).

The above result without the use of p-function was originally proved by Shelah[17] for", =w. Velleman used his simplified ("" I)-morasses to get the above resultwithout the use of p-function.

Theorem 5.5. (Todorcevic; see Bekkali [2}.) Using p one can obtain a canonicalHausdorff-gap.

Theorem 5.6. (Todorcevic; see Bekkali [2}.) Using p and assuming OWl one candefine a function with the Ll-property by

!(a,{3) = {~< a : p(~,a) :::; p({3,a)}.

References

1. J .Baumgartner and S.Shelah, Remarks on superatomic Boolean algebras, Ann.Pure . Appl. Logic 33 (1987) 109-130.

2. M. Bekkali, Topics in Set Theory, (Notes from lectures by Stevo Todorcevic), Lecture Note in Mathematics, 1476, Springer-Verlag 1991.

3. K. Eda, G. Gruenhage, P. Koszmider, K. Tamano and S. Todorcevic, Sequentialfans in topology, Topology Appl. 67 (1995), no. 3, 189-220.

4. I. Gorelic, The Baire category and forcing large LindelOf spaces with points G6,

Proc. Amer. Math . Soc. 118 (1993) , no. 2, 603-607.5. R. Jensen, 0 implies GCH, handwritten notes; see [6] .6. R. Jensen and K. Szlechta, Results on the Generic Kurepa Hypothesis; Arch. Math .

Logic 30 (1990) 13-27.7. I. Juhasz and W. Weiss, On thin-tall scattered spaces, Colloq. Math. 40 (1978/79),

no. 1, 63-68.8. P. Koszmider, Etude is simplified morasses, Seminar notes, Toronto.9. P. Koszmider, Kurepa trees and topological nonreflection, Preprint.

10. P. Koszmider, Splitting ultrafilters of the thin-very tall algebra and initially Wt compactness, Preprint

11. P. Koszmider; On the existence of strong chains in p(wt)/Fin, Journal of SymbolicLogic, to appear.

12. P. Koepke and J. C. Martinez , Superatomic Boolean algebras constructed frommorasses, J. Symbolic Logic 60 (1995), no. 3, 940-951.

13. C. Morgan, Morasses, square and forcing axioms; Annals of Pure and Applied Logic80 (2) (1996) 139-163.

14. C. Morgan, Preprint.15. M. Rabus, An w2-minimal Boolean algebra, 7h1ns. Amer. Math. Soc. 348 (8)

(1996) 3235-3244 .16. S. Shelah, On some problems in general topology, in: Set theory (Annual Boise

extravaganza in set theory (BEST) conference, 1992/1994, Boise State University,Boise, Idaho) , T. Bartoszynski et aI., (eds.), Contemp. Math . 192 (1996) 91-101.

17. S. Shelah, Can you take Solovay's inaccesible away? Israel J . Math. 48 (1984) 1-47.

98 PIOTR KOSZMIDER

18. S. Todorcevic, Partitioning pairs of countable ordinals; Acta Mathematica 159(1987) 261-294.

19. S. Todorcevic, Remarks on Martin's Axiom and the Continuum Hypothesis, Ganad.J. Math . 43 (1991), no. 4, 832-851.

20. B. Velickovic, Forcing axioms and stationary sets ; Advances in mathematics 94, No2 (1992) 256-284.

21. D. Velleman, Simplified morasses, J. Symbolic Logic 49 (1984), no. 1, 257-271.22. D. Velleman, On a combinatorial principle of Hajnal and Komjath, J. Symbolic

Logic 51 (1986), no. 4, 1056-1060.

MODELS AS SIDE CONDITIONS

PIOTR KOSZMIDER1

Department of MathematicsAuburn UniversityAuburn, AL 36849-5319USA

1. Introduction

The point should be in building new combinatorics which will enable us to constructnew posets and discover new reasons for the existence of generic conditions.

S. Todorcevic: Partition Problems in Topology

The purpose of the method in the title, which was established by Stevo Todorcevic([4], [5]), is to overcome certain problems in proving that some partial orderspreserve WI' More specifically, the problems with amalgamating conditions whileattempting to prove the countable chain condition of partial orders. The generalidea is to put inside conditions of forcing notions some extra demands which willmake the amalgamation easier . The point is that if one involves models into theseextra demands, one is rewarded. These extra demands increase the complexity ofthe condition (and are no longer finite) and so one has no choice but to forgetabout proving that the modified partial order satisfies the c.c.c. One tries to provethe properness of the modified forcing. This boils down to the proof that certainconditions are generic/ over certain models.Norking with proper forcings one encounters very specific technical problems, asin the case of the work with c.c.c. partial orders. There are many methods for

lThe author waspartially supported by the NSF ofUSA Grant DMS-9505098. The author was also partially supported during the conference by Centre de RecercaMatematicafor whichhe is very grateful to the organizers of the conference: Joan Bagaria and AdrianMathias.

2Recall that if a partial order P is fixed and a countable elementary submodel M ofsome H" is given, then q is said to be a (P, M)-generic if and only if D n M is predensebelow q for every dense D which is is an element of M. This notion has been developedby Shelah. See Martin Goldstern's paper for the discussion of this notion.

99


100 PIOTR KOSZMIDER

proving the c.c.c., e.g. the fundamental one is the use of the ~-system lemma, butwe have totally different arguments for proving the c.c.c. of the measure algebraor the forcing for adding a dominating real etc. Also there are many refinementsof the method that uses the ~-system lemma: Baumgartner's use of a uniformultrafilter for the forcing specializing Aronszajn trees, the use of p-function or afunction with the ~-property etc. In case of proper forcings we have much lessmethods available.

We should note that the present situation of our understanding this point is extremely unsatisfactory. For example, one can hardly find in today's literature twodistinct combinatorial arguments which are finding a condition and proving itsgenericity with respect to a given countable elementary submodel.

S. Todorcevic; Partition Problems in Topology

One of the methods for proving that a condition q is a (P,N)-generic is Todorcevic's method which is the subject of this paper. Suppose we added extra demandsto our finite conditions and we made amalgamations easier, but now amalgamations are apparently irrelevant to properness. This is not true, amalgamations mayplay the crucial role in the proof that a certain condition is a generic. To see this,translate a proof of the c.c.c, into the context of generic conditions:Suppose that we want to prove that a forcing notion P, made of finite conditions,satisfies the C.C.C. We suppose A = {P{ : ~ < Wi} is an uncountable collection ofelements of P. Now we can find a countable M ~ H K for large enough", such thatP,A E M . Now take q E A-M. We can talk about Mnq = qlM as the conditionsof P are finite.

We haveHK F 3q' E A qlM ~ q' & q' is such and such.

because q witnesses this sentence, so by the elementarity

M F 3q' E A qlM ~ q' & q' is such and such.

(again, as qlM is finite it is in M). So take such a q' = q(M)Now both q(M) and q are such and such and they agree on the intersection,so it may be easy to amalgamate them. Actually this argument may replace allapplications of the ~-system lemma or the Pressing Down Lemma or any thinningdown of the original A. Even Alan Dow, proves the ~-system lemma that way (see[3]), one even gets here a stationary ~-system (going in this direction one arrivesat the ~-system lemma for nonspecial trees (see [6])) .Now let us present a general scheme of using amalgamation in the proof of thegenericity of a condition over a model. The proof mimics the above scheme of theproof of the C.C.C

Let P,M,p be given such that PEP; M ~ HK ; p,P E M. We want to constructp. EPa (P, M)-generic .

1. Define p. (so that the following steps can be performed)

MODELS AS SIDE CONDITIONS 101

2. Take q :5 p* and D E M which is dense and open in P; and without loss ofgenerality, assume that qED. (We want to get sllq, s E M n D, so if we gofrom q to ql :5 q such that ql ED and we do it for ql, s will work also for q,so that's why we can without loss of generality assume that qED)

3. Get a restriction qlM and use the elementarity of M to get a projection q(M)of q on M (q(M) must be in M) .

4. Amalgamate q and q(M) into s. Now s witnesses that D n M has a conditioncompatible with q (the condition is q(M)).

The main idea of the method of models as side conditions is to construct a forcingnotion whose conditions p are of the form p = (Dp,Np) where Dp is the workingpart and comes from the original forcing with finite conditions and N p is an Echain of countable elementary submodels of some H>. . It turns out that if wework with this modification, many points of the above scenario of the proof of thegenericity of a given condition over a model M get simpler in a canonical way.In the rest of this paper we discuss two standard situations when using modelsas side conditions helps us overcoming problems in executing the above scenario.Note also that if no problems occur while executing the above scenario for forcingwith finite conditions, then the forcing just satisfies the C.C.c.One must note now that if submodels of HK, are now parts of conditions of ourforcing, we are facing the proof of the (P,M)-genericity of p* for M which is acountable elementary submodel of H>. for A much bigger than K.. Here it will becrucial to note that if K. E M -< H A, then M n HK, -< HK,. This can be easilychecked. This fact allows us to think about side conditions as about the actualmodel M over which we have to prove the genericity.

The actual definition of the conditions is designed to make the definition of (M, Q)generic conditions a triviality.

Alan Dow;An Introduction to Applications of Elementary Submodels to Topology

The crucial part of any of our definitions will be the "side conditions" determininghow Dp and Np interact. The side conditions are usually obtained by a fine analysisof all reasons which prevent the working part of a given condition to be genericover a given countable elementary submodel M (of much larger structure than H A)

over which we would like p to be generic.S. Todorcevic; Partition Problems in Topology

102 PIOTR KOSZMIDER

2. First use of models as side conditions: the possibility of restrictingconditions to models.

Consider a forcing P consisting of finite functions from subsets of WI into WI withthe inverse inclusion as the order. The forcing is not c.c.c., it collapses WI. Theproof of the properness along the points 1-4 from the introduction would breakdown when we considered qlM. Say, e.g., M n WI = 8, a E M - dom(P) andq(a) > 8, then the natural restriction of q to M as a function is not in M, and sothe elementarity in 3) cannot be used.Let us modify the forcing by adding models as side conditions. The intention is tomake sure that the restrictions belong to the models over which we want to getgeneric conditions. Here side conditions mimic these models. So define a modifiedpartial order P' consisting of pairs (f,N) where the working part f is in P andcomes from the original forcing with finite conditions and Np is an E-chain ofcountable elementary submodels of some H~. Additionally explicitly demand theexistence of restrictions:

"IN EN Va E Nndom(f) f(a) E N

Now according to 1. define the generic p. as (fp,NpU {M n H~}), take q ~ p. ina dense DE M and now fqlM E M, so one can use the elementarity of M to getq(M) extending qlM. The functions are compatible as fqlM is included in both ofthem. Appending models as side conditions paid off. There is one problem though,we need to have also a restriction to M of our new side part of the conditionq. This is the reason why we considered E-chains not just single models. Namely,NqnM E M because MnH~ E NqandNq is an E-chain. We put NqlM =NqnM.

A nice interaction of models from Nq, NqI M' Nq(M) 'Everything due to the fact that Nq is an E-chain containing M n H ~ .

One needs to note that even in situations when models as side conditions are notappended to get restrictions, once the models are added we are facing the problemof getting restrictions of the elementary chains of models. In this sense this functionof side conditions as providers of restrictions is present in any instance of themethod. The above example was provided just for expository purposes as this partof the argument appears in any use of the method (although one can add a clubwith finite conditions a very similar way. This club as well as the function addedby the above forcing have many interesting properties) .

3. Second use of models as side conditions: decomposing projections ofn-tuples to n-projections of singletons.

Let us first see an example of this use and then let us discuss its general context.We will consider killing S-spaces by proper forcings. Recall that X is an S-spaceif and only if it is a regular topological space which is hereditarily separable (eachsubspace is separable) but it is not hereditarily Lindelof (there is a subspace whichdoes not have the Lindelof property). To see all possible contexts of the issue see[7]. We want to kill S-spaces to get the consistency of the nonexistence of S-spaces(due to Todorcevic). Suppose X is an S-space. By going to a subspace, w.l.o.g. onecan assume that it is not Lindelof and so there is an open cover U of X which hasno countable subcover. Using this fact one can construct by induction sequences{xa : a < wd ~ X and {Ua : a < wd ~ U such that

Va E WI X a E Ua and V{3 < a < WI X a ¢ Up

x

For future purposes, using regularity of X, refine Ua's so that

V{3 < a < WI X a ¢ Up

To kill the S-space X, it is enough to add an uncountable B ~ WI such that

Va < {3 < WI a, {3 E B X a ¢ Up

Indeed, in that case we simply get that X a ¢ Up whenever {3:/= a and now we arenoting that {x a : a E B} is an uncountable discrete space (i.e., points are open

104 PIOTR KOSZMIDER

in the subspace topology (this is witnessed by UQ 's)), and so it is not separable,contradicting hereditary separability of X. So adding B as above would kill thefact that X is an S-space. To get B as above, it is is natural to force with a forcingP consisting of p E [WI]<w such that

'Va,f3 E p, a < f3 X Q ¢ Up

As the conditions are finite, it is reasonable to hope that the forcing satisfies thec.c.c. To prove this, take an uncountable A ~ P, take an elementary, countableM -< HI(, and q ¢ A - M, define qlM = q n M and try to find q(M) E An Mwhich is compatible with q.Now we will note that in a simple case, when q - M = {xe} for some ~, we canfind q(M) as required. Then we will see what kind of problems we are facing whenconfronted by q - M of size bigger than one and trying to repeat the tricks of theprevious case. Finally we will describe how to overcome the problems by addingmodels as side conditions.Note that for each countable Y E M the following sentence is witnessed by ~

HI( 1= There is 1] ¢ Y such that qlM U {71} E A

soM 1= There are uncountably many 1] such that qlM U {71} E A

Now q(M) is going to be of the form qlM U {71} such that q(M) E A and q(M) iscompatible with q. For the later we need 1] ¢ Ue . Here is how we find it: ConsiderT = {1] E WI : qlM U {71} E A} (we proved above that T is uncountable). Notethat T EM. Now use hereditary separability of X and elementarity of M to finda countable S ~ T such that S ~ M and S is dense in T . Now take 8 E T - ~

(using the uncountability of T) and use the fact that X6 ¢ Ue and regularity of Xand the density of Sin T to get 71 E S close to X6, i.e., such that xfJ ¢ Ue. Thiscompletes the search for appropriate 71 and q(M).Now let us see what kind of problems we are facing if nEw, the size of q - M ={Xel' ...,xen} is bigger than one.Trying to mimic the above argument, we define T ~ [x]n by

T ={ffE [wd n: qlMUffE A}

As before T EM, now we want to find ff E T such that

ffn(Uel u ...UUeJ =0

To mimic exactly the previous proof we would need hereditary separability of xn!This property is much stronger than hereditary separability. This is the momentwhen we are stuck in the proof of the c.c.c, Of course one can assume that X" ishereditary separable. Such spaces are called strong S-spaces. In that case we notethat we have proved the following theorem.

Theorem. (K. Kusien) MA implies that there are no strong S-spaces.

But what about the consistency of no S -spaces? (see the paper [1] for first countable S-spaces (clearly not strong S-spaces) compatible with MA).Now to overcome the problem, we are going to add models as side conditions.This will allow us to use n-times a property of X instead of using a property ofX" , How can models be used for this? (compare with the closed unbounded trickdue to Shelah (see [2])). Suppose q - M = {6,6}. To be successful with findingappropriate q(M), we would define

T0 = {171 E WI : 3172 qlM U {171 , 172} E A}

As before we can find 171 E M n T0 such that 171 rf. Ue! U Ue2.Now we define

TTll = {172 E WI : qlM U {171,172} E A}If we knew that TTl! was uncountable we would apply hereditary separability of Xto TTll and we would get as before 172 rf. Uel U Ue2·But there is no reason to believe TTl! has more that one element. Now assume thatthere was a countable model N -< Hn such that ~1 E Nand 6 rf. N. In that casewe may consider T's redefined by

T0 = {171 E WI : 3172 qlM U {171, 172} E A 3 N -< s, 171 EN, 172 rf. N}

Now if M -< H>. for .A big enough we have 171 E Tn M such that 171 rf. Ue! U Ue2.Now TTl! unlike before, is uncountable because

for every countable YEN, where N is the model from definition of T0 witnessingthat 171 E T0'So now, let us try to modify the forcing demanding separation of elements bymodels.

We shall always have the requirement that Np separates Dp with the meaning thatfor every two distinct x and y in Dp there is an N in Np containing exactly oneof them. Note that uniquely determines a linear ordering <p on Dp : x <p y iff3N E Np (x E N & y rf. N).

S. Todorcevic; Partition Problems in Topology

Consider P' consisting of conditions p = (ap,Np) such that

a) ap E P,

b) Np is an elementary E-chain of countable submodels of H>.,

c) For every 0, (3 E ap , if 0 < (3, then there is N E Np such that 0 E Nbut (3 rf. N.

Now let us prove that P' is proper following the scenario from the introduction.Take M -< H>. containing p and all relevant objects. Define

p* = (ap,NpU {M n Hn } )

106 PIOTR KOSZMIDER

Now take q ~ p* and D E M dense in P. W.l.o.g. assume that qED. Restrict qto M by

qlM = (aq n M,.Nq n M)

Let n = laq - MI . Now define T in M by

Now one can use the separation as above to get T' ~ T such that every node of T'splits uncountably (when viewed as a tree). This enables us to perform the trickfor singleton n-times to get the appropriate n-tuple.Note that the passage from singletons to n-tuples is not just a technicality. It isfundamentally new quality. Many notions are defined in terms of sets of singletons,e.g., irredundance of Boolean algebras or incomparability in partial orders. If wewant to force certain substructures of countably irredundant Boolean algebrasor partial orders without uncountable antichains, we cannot use correspondingproperties of powers. Models as side conditions can be used in the above contextsin similar way to the use in the case of S-spaces. In particular S. Todorcevicproved that PFA implies that there are no S-spaces (see [6]) , every uncountableBoolean algebra has an uncountable irredundant set [8] , every partial order withoutuncountable antichains is a union of countably many directed sets [4].

4. More on the method

One can say that the method of models of side conditions allows us to performall the tricks with proper forcing easily and in one blow (compare the closed andunbounded trick or others in [2]) . To other standard uses of the method one couldinclude avoiding bad sets:

Suppose we want to force an uncountable subset A of a given structure E with Ahaving some specific properties . The natural thing would be to force with the posetP of all finite approximations to A. We would like to prove that P , or a certainsubposet of it, is proper. So let N -< Ho be countable such that p, PEN. let q ~ pbe a condition for which we would like to prove that it is (N, P)-generic. So letDEN be dense open and let r ~ q. In most of the cases we shall not be able toshow that there is an sED nN such that rlls, since r - N will be in certain "bad"places with respect to N. To avoid this, we shall simply "add" N to be a "sidecontiition ", saying explicitly that r - N is not in a bad place with respect to N .

S. Todorcevic; A Note on Proper Forcing

It is also not difficult to see that a generic set in a forcing with models as sideconditions gives an elementary chain of countable sets coveringthe entire structurethat we are working with. Thus the size of the structure is collapsed to WI ' Thisis o.k. when we are working with PFA.


None of the statements in this section needs large cardinals for its consistency.This may not be so clear for the last three statements since the side conditionsobviously do collapse cardinals.

S. Todorcevic; Partition Problems in Topology.

To avoid collapsing cardinals one needs to give up elementary chains and replacethem with more complicated structures (see [5]). See the paper of Max Burkein this volume to look at many consistency results which can be obtained usingmodels as side conditions. They can be obtained directly using the method inthe context of PFA, or axiomatic consequences discussed by Max can be used forconcluding them.

References

1. U. Abraham and S. Todorcevic, Martin's Axiom and first-countable S- and L-spaces,in: Handbook of Set Theoretic Topology, 2nd printing, (K. Kunen and J.E. Vaughaneds.), North-Holland and Elsevier Science Publishers, Amsterdam and New York,(1984) 327-346.

2. J .Baumgartner, Applications of proper forcing, in: Handbook of Set Theoretic Topology, 2nd printing, (K. Kunen and J.E. Vaughan eds.), North-Holland and ElsevierScience Publishers, Amsterdam and New York, (1984) 913-959.

3. A. Dow, An introduction to applications of elementary submodels to topology,Topology Proc. 13 (1988) 17-72.

4. S. Todorcevic, A note on the proper forcing axiom, in: Axiomatic set theory (Boulder, Colo., 1983), Contemp. Math. 31, Amer . Math. Soc., Providence, R.I., (1984),209-218.

5. S. Todorcevic, Directed sets and cofinal types, TI-ans. Amer. Math. Soc. 290, no. 2,(1985) 711-723 .

6. S. Todorcevic, Partition relations for partially ordered sets, Acta Math. 155 no. 1-2,(1985) 1-25.

7. S. Todorcevic, Partition Problems in Topology; Contemporary Mathematics 84;AMS, Providence, Rhode Island, 1989.

8. S. Todorcevic, Irredundant sets in Boolean algebras, 1rans . Amer. Math. Soc. 339(1993), no. 1, 35-44.

AN ORDINAL PARTITION FROM A SCALE

JEAN A. LARSONIDepartment of MathematicsUniversity of FloridaGainesville, FL 92611USA

Dedicated to the memory of Eric Milner

Abstract. In this paper, some theorems of Hajnal are revisited to weaken thehypothesis of CH to the existence of a short scale (ll = NI). In particular, ifII = NI , then there is a triangle-free graph G on WI • W such that every subsetX ~ WI • W of order type WI • W has an edge, and there is a triangle-free graph Gon WI 2 so that every subset X ~ WI 2 of order type WI 2 has an edge.

In the notation of Rado, these results may be expressed as

II = NI ==> WI'W ft (WI ' W , 3 )2 and

II =NI ==> WI2 ft (WI

2,3)2.

These results are generalized to partition relations for K, • ). and K,2, when ). isregular and K, = >.+.

1. Introduction

Ramsey's Theorem has been generalized in many different ways. In this paper, weconsider the partition calculus for ordinals: the generalization to partitions of finitesubsets of ordinals in which we look for homogeneous sets of specified order type .R. Rado's arrow notation, first introduced in [11], encapsulates such statementsin a compact form. Write a ~ ({3, m)2 if for every ordered set S of order type a

and every graph on S, either there is an independent subset X ~ S (one omittingall edges of the graph) of order type {3, or there is a complete set Y ~ S (one inwhich all pairs are joined by edges of the graph) with WI = m. Write a ft ({3,m)2

if the negation of the above statement holds.

IThis work was partly supported by grant number DMS9306286 from the NationalScience Foundation.

109

C.A. Di Prisco et al. (ed.), Set Theory, 109-125.© 1998 Kluwer Academic Publishers .

110 JEAN A. LARSON

This introduction summarizes special cases of partition relations of the forma ~ «(3, m)2 when a and (3 are both infinite and a = K, • A is the product of twocardinals K, ~ A. Two of these results are strengthened in subsequent sections.

First consider products a = K, • l for finite i. Erdos and Rado proved in their1956 paper [12] that W· l ~ (a,l)2 for finite l and a < w ·2 and furthermore,that w · l -lt (w + l,l + 1)2. Moreover, they showed that w·l ~ (w· n,m)2 if andonly if l ~ r(K:, Lm ) where r(K:, Lm ) is least natural number p so that everydirected graph of order p either has an independent set of n vertices or contains atransitive tournament of order m . In their 1967 paper [13], they asked the questionof whether or not the same equivalence holds for uncountable K, in place of w, andBaumgartner in [1] showed that it does. See [18] for a survey of known resultsabout the digraph Ramsey number r(K:, Lm ) .

Next consider a = K, • A where K, > A ~ w. P. Erdos and A. Hajnal, in their1971 paper [9], proved a positive result for the special case of m = 3.

Theorem 1.1 (Erdos and Hajnal [9]). If K, is a successor cardinal and A< K"

then K, • A~ (K,. <5,3)2 for all <5 < A.

Shelah and Stanley used a cardinality argument in their 1987 paper to provea positive result for A=w that works for all m < w, not just m =3.

Theorem 1.2 (Shelah and Stanley [20]). For all regular K" if A < K, impliesA~o < K" then K, • w ~ (K, • w, m)2 for all m < w.

In particular, if CH holds, then W2 ·W~ (W2 ·w,m )2 for all m < w. This resultanswers a question of Erdos and Hajnal [9] who proved that K,·W ~ (K, ·w, m)2 forall m < w when K, is a singular cardinal cofinality w.

Shelah and Stanley proved a related result for products K, • A where K, > A > win their 1993 paper.

Theorem 1.3 (Shelah and Stanley [21]). (Assume GCH). If m > 2, w{3 isregular and (3 > a + m - 2, then W{3 . Wa+m-2 ~ (w{3 . wa, m)2.

The proof uses Hajnal's free subset result and the fact that there are at mostNa+m - 2 subsets of Wa+m-2 of power < Na+m - 3 . Note that under GCH, if m =3,K, = w{3 is a successor and K, > A = WaH, then Theorem 1.3 is a special case ofTheorem 1.1.

Attention has been focused on partition relations of the form K,. A~ (K, .A, 3)2.The theorem below gives a strong negation of K, • A ~ (K, • A,3)2 in the case

K, = A+ =2'\ (see [2] for the attribution).

Theorem 1.4 (Hajnal [9]). If K, = A+ = 2>' and A is regular, then K, • p -It(K, • A,3)2 for all infinite cardinals A and ordinals p < K, .

In Section 2 of the current paper, the above theorem is strengthened by replacing the hypothesis that K, = A+ with the assumption there is a scale undereventual domination of length K, in xA. For the case of A = w, this assumptionis equivalent to the assumption that the dominating number tJ is equal to N1• Inprivate conversation, Jorg Brendle wondered whether this assumption could beweakened to b = tJ, that is, the assumption that there is a scale (not necessarilyshort).

AN ORDINAL PARTITION FROM A SCALE 111

In a paper which appeared in 1987, J . Baumgartner and A. Hajnallooked atsuccessors of singular cardinals.

Theorem 1.5 (Baumgartner and Hajnal [4]). If J.L+ = 2J.l and cf J.L = A, thenJ.L+ • AIt (J.L+ . A, 3)2.

One can easily extend this result to show under the same hypotheses thatJ.L+ • P It (J.L+ • A,3)2 for all p < A+. See the end of the next section for theargument. One can use pinning and the result above to generalize Theorem 1.4to all successor cardinals K, = J.L+ = 2J.l, not just successors of regular cardinals.It is an open question whether the hypothesis J.L+ = 2J.l can be replaced by theexistence of a scale of length J.L+ in cf J.l J.L .

However,Baumgartner has shown that the partition relation WI 'W -t (WI'W, 3)2cannot be settled in ZFC.

Theorem 1.6 (Baumgartner [3]). If Marlin's Axiom for NI holds, then WI .W-t (WI · w,m )2 and WI ·w2 -t (WI ·w2,m)2, for all m < w.

As a companion to their CH result that W2 . W -t (W2 . W, m)2 for all m < w,Shelah and Stanley prove the following consistency result:

Theorem 1.7 (Shelah and Stanley [20]). If ZFG is consistent, then so is ZFG+ -,GH + W2' wit (W2 ' W,3)2.

They remark that the forcing construction generalizes to K,++ • K, for regularuncountable K, which satisfy A+ < K, ==> 2A < K,; and they comment that thegeneralized construction uses conditions of size less than K,. Since the continuumis c = N2 in this model, this model and, by Theorem 1.4, models of CH, all showthe consistency of c . W It (c . W, 3)2.

Question 1.8. Is it consist ent that c . W -t (c . W, 3)2?

A cardinal K, is weakly compact if K, -t (K" K,)2 • Baumgartner has shown how to

lift certain partition relations to products with a weakly compact cardinal.

Theorem 1.9 (Baumgartner [17]) . If K, is weakly compact, 0 < K, and 0 -t(0, m)2 , then K, • 0 -t (K, • 0, m)2 .

Since the Erdos ; Dushnik, Miller Theorem [7] says that for any infinite cardinalA, one has A-t (A,No)2, it follows that K,' A-t (K,. A, m)2 for every weakly compactcardinal K" every cardinal A < K, and every finite m.

Since W2 is the double successor of w, another natural product to consider isW3 -WI·

Theorem 1.10 (Shelah and Stanley [20]).(i). If ZFG is consistent, then so is ZFG + GGH + W3 . WI It (W3 . WI , 3)2.(ii). If ZFG + "there exists a weakly compact cardinal" is consistent,

then so is ZFG + GGH + ("1m < w) (W3 ' WI -t (W3 · wI,m )2) .

Shelah and Stanley comment in [21] that "we do not currently know how,even starting from a measurable cardinal , for example, to produce a model where2N1 > N2 and w3 WI -t (W3 WI , 3)2" .

Hajnal and later Baumgartner used recursion and the cardinality condition2A = ,x+ to build examples of ,x+ . A It (A+ . A,3)2. In this paper, a short scale

112 JEAN A. LARSON

is used. Lee Stanley and Dan Velleman and independently, Charles Morgan [23],building on work by Miyamoto [19] in his thesis, used a morass.

Theorem 1.11 (L. Stanley and D. Velleman; C. Morgan [23]).If2N1 =N2 and there is a simplified(W2, I)-morass with linear limits, then W3 'WI f+(W3· WI,3)2.

Corollary 1.12 (L. Stanley and D . Velleman; C. Morgan [23]).If 2N1 = N2 and W3 • WI --+ (W3 • WI, 3)2, then either N2 or N3 is inaccessible in L.Thus if ZFC + 2N1 =N2 + W3 • WI --+ (W3 • WI, 3)2 is consistent, then so is ZFC +"there exists an inaccessible cardinal".

Finally turn to partition relations ofthe form K,2 --+ ({3, m)2. The first result ofthis form dates back to the 1950's.

Theorem 1.13 (E. Specker [22]). For all m < W, w2 --+ (w2 , m)2.

The case for triangles is particularly nice.

Theorem 1.14 (Erdos and Hajnal [9]). If K, is a successor cardinal and>. <K" then K,2 --+ ({3,3)2 for all {3 < K,2.

However, the situation changes when one asks for more than a triangle.

Theorem 1.15 (J. Baumgartner and A. Hajnal [4]). If >. is regular andK, =>.+ =2\ then K,2 f+ (K,. >.,4)2 .

It would be interesting to know if the hypothesis >.+ = 2>' can be weakened tothe existence of a short scale in x>.. To show that the hypothesis >.+ = 2>' is notnecessary, they also prove the following theorem:

Theorem 1.16 (J. Baumgartner and A. Hajnal [4]). AssumeV 1= (A = cf A ;::: W & GGH&c£J1- > A) . There exists a A-complete notion of forcingP satisfying the u-c.c. such that

There is a misprint in the statement of this theorem in [4]. There they conclude"V P F 21' = K, & K, . >.+ f+ (A+ . A, 4)2." However, their forcing adds a graph whichhas no edges between any pair of elements from the same fiber and each fiber hascardinality K,. Thus if in the extension, K, > >.+, then a single fiber provides anindependent set of size K, > >.+ .>., contrary to the statement as it appears in theirpaper.

Finally consider the special case K,2 --+ (K,2, 3)2.

Theorem 1.17 (Hajnal [15]). For all regular cardinals >', if K, =>.+ =2>' I thenK,2 f+ (K,2, 3)2.

Section 3 strengthens Theorem 1.17 by replacing the hypothesis that K, = >.+with the assumption there is-a scale under eventual domination of length K, in x>..Baumgartner generalized Theorem 1.17 to successors of singular cardinals.

Theorem 1.18 (Baumgartner [2]). For all cardinals >', if K, = >.+ = 2>., thenK,2 f+ (K,2, 3)2.

In addition, he showed that for a singular strong limit cardinal K" one could usecanonization to reduce the question for K,2 to the question for 7 2 where cf K, = 7 .

Theorem 1.19 (Baumgartner [2]). Ij « is a strong limit cardinal, then ",2 -t(",2, m) if and only if (cf ",)2 -t (fcf ",)2, m)

It is known that if", is weakly compact, then ",2 -t (",2, m)2 for all m < w. See[14], for example.

Baumgartner [2] has also shown how to take a x-Souslin tree on a regularcardinal '" and use it to build a witness to ",2 f7 (",2,3)2. He concluded that if '"is regular and ",2 -t (",2,3)2 holds, then the x-Souslin Hypothesis also holds . Theconverse need not hold, since Jensen [5] has a model in which not only the WI

Souslin Hypothesis (SH) holds, but also CH holds, so W1 2 f7 (WI 2,3)2 by Theorem1.17.

Jensen [16] has shown under the assumption of GOdel's Axiom of Constructibility (V=L), that '" is weakly compact if and only if the x-Souslln Hypothesis holds.Hence, under the assumption of V=L, all is known about ",2 -t (",2,3)2.

Theorem 1.20 (Baumgartner [2]). If V = L, then for all infinite cardinals «,",2 -t (",2 ,3)2 if and only if cf '" is weakly compact.

Baumgartner drew attention to several open questions in [3] . In particular, thefollowing question remains open.

Question 1.21. Is it consistent that W1 2 -t (W1 2, 3)2?

The set theoretic notation of this paper is fairly standard, and some is reviewedin the next section which discusses eventual domination, scales, ladders, graphs onproducts of ordinals, fibers and sets up notation for the rest of the paper.

2. The example for '" . A

The set V = A x '" has order type '" . A under the lexicographic ordering, whichwill be denoted < for simplicity. It will be the underlying set of vertices for theexample built in this section to witness x . A f7 (",. A,3)2, under the hypothesesthat A is regular and there is a scale of length '" in AA.

Since V is linearly ordered, any edge has a natural orientation. Such an orientation allows us to make the next definition .

Definition 2.1. Suppose H = (U,F) is a graph on a linearly ordered set of vertices U. The out-neighborhood of a point u E U in H is the set

nhbt(u) := {v E U I {u, v} E F &u < v}.

The subscript H will be omitted when the graph is understood.

Notice that any graph on V has the property that each edge {x, y} is in exactlyone out-neighborhood. The global strategy for defining the desired graph on V isto work by recursion, and at stage (3, define the sets nhb+(x) for all x E A x {{3} .To simplify the task, all edges are assumed to move inward.

Definition 2.2. For any ordinals a and p, say {(J.l,{3) , (v,'Y)} E [a x p]2 movesinward if either (J.l < t/ < 'Y < (3) or (v < J.l < (3 < 'Y).

114

(p, (3)

JEAN A. LARSON

. . . . . . . . . . . . . . . . . (v, (3)

(v,'Y)

(p, p)• I

(v, v)

(v,p)

The edge {(m,[3), (n,'Y)} moves inward

The next lemma, which follows from the definitions, indicates that to definenhb+(x) for x = (m,[3) E V, one need only look at points of AX [3.

Lemma 2.3. For any ordinals a and p, and any gmph (a x p,F), if all edgesmove inward, then nhb+(p,[3) ~ a x [3 for all (p,[3) E a x p,

To build an example which witnesses K. • A fi (K. • A, 3)2, there are two basictasks: (1) make sure large sets X ~ V have edges of the graph; (2) make sure thereare no triangles.

A brute force approach is used to avoid triangles . Since all edges move inward,if there were a triangle, it would be of the form { (m, (3), (n, ,), (P, 0) } where m <n < p < 0 < 'Y < (3; in such a case, {(n,'Y) , (P,a)} S;; nhb+(m,,8). Thus amonggraphs in which all edges move inward, the triangle-free ones are exactly those inwhich all out-neighborhoods are independent.

Lemma 2.4. For any ordinals a and p, and any gmph H = (a x p,F) in which alledges move inward, the gmph H is triangle-free if and only if nhb+(p,,8) ~ a x [3is independent for all (p,,8) E a x p,

The notion of fibers is introduced to make it easier to talk about large subsetsofV.

Definition 2.5 . For any ordinals a, p and any set A ~ a x p, call AI3 = A n({13} x p) the 13th fiber of A.

A set X C V has order type K.. Aif and only if it has Amany fibers of cardinalityK.. Translate this statement into one about limit points.

Definition 2.6. Suppose Y ~ a x p, 0 ~ p is a limit ordinal, and "1 < a . Then 0is a limit point of Y'1 if 0 is a limit of { 'Y I ("1, 'Y) E Y}.

Thus a set X C V = A x K. has order type K. • A if and only K. is a limit pointof X p for A many fibers X p• This statement can be reflected down to 0 < K. withcfO= A.Definition 2.7. Suppose X ~ Ax K.. Then 0 < K. is a multiple limit point of X ifcf a= A and there are Amany p < Afor which 0 is a limit point of X p •

Lemma 2.8. Suppose that K, and A are cardinals with cf K, > cf A = A. For anysubset X ~ A x K, of order type K, • A under the lexicographic ordering, there is alimit ordinal § < K, so that § is a multiple limit point of X .

Proof. Under the hypothesis of the lemma, A many fibers of X have cardinalityK,. Define a sequence of pairs (Pa,6a) by recursion on 0 < A. Let Po be the leastordinal P so that X p has cardinality K, and let 60 = A. If (P/3' 6/3) have been definedfor 0 $ [3 < 0, let Pa be the least P > U/3<a P/3 so that X pa has cardinality K,.

Since A is regular, such a value can always be found. Let 6a be the least ordinald > U/3<a 6/3 so that for all [3 with 0 s [3 s 0, there is some (P/3,7) E X with6/3 < 7 < d. Let P := {Pa I0 < A}, D:= {§a 10 < A} and let 6 = supD. SinceK, has cofinality K" it follows that 6 < K,. Moreover for all pEP, 6 is a limit pointof X p , so it is a multiple limit point of X . 0

With this terminology, the strategy for avoiding large independent sets may beroughly outlined.

Lemma 2.9. Suppose that K, and A are cardinals with cf K, > cf A = A. For anygraph H = (A x K" F) , let 'Po (H) be the following statement: for every limit ordinal§ , every set Y ~ Ax 6 for which 6 is a multiple limit point, and every J-L < 6, thereis some b(6,J-L, Y) so that nhb+«J-L, [3)) n Y ::I 0 for all [3 with b(6,J-L, Y) $ [3 < K, .

For any graph H = (A x K" F), if 'Po (H) , then H has no independent sets of ordertype K, • A.

Proof. Suppose H = (V, F) is a graph on V = A x K, satisfying the hypothesis ofthe theorem. Further suppose X ~ V is an arbitrary set of order type K, • A. ByLemma 2.8, there is a limit ordinal 6 which is a multiple limit point of X. LetY =X n (A x 6). Then 6 is also a multiple limit point of Y . Use the fact that Xhas A many fibers of cardinality K" to find a least ordinal J-L so that the fiber X /l

has cardinality K, . Let b=b(6, p, Y) be the lower bound posited by the hypothesisof the lemma. Since XI' has cardinality K" there is some /3 > b = b(8, p, Y) so that(J-L, /3) EX. By hypothesis, nhb+«J-L, /3)) n Y ::I 0, so the set has some element(V,7). Thus {(J-L, /3), (v, 7)} witnesses that X is not independent. Since X wasarbitrary, the lemma follows. 0

The need to add edges to guarantee that large sets include an edges must be balanced with the need to avoid adding edges to guarantee that the out-neighborhoodof every point is independent. To that end , another simplifying hypothesis is made :the out-neighborhood of each point has small fibers.

Definition 2.10. For any ordinals 0, P = K" a set Z ~ 0 x P has small fibers ifall fibers ZII have cardinality less than A.

Of the four simplifying inductive hypotheses described above , three are explicitand well-tailored to a definition by recursion: (1) all edges move inward; (2) theout-neighborhood of every point is independent; (3) the out-neighborhood of everypoint has small fibers ; In his construction, Hajnal used (1) and (2), and made astronger hypothesis (3)' that each point (J-L, {3) is joined to at most one point ofany {p} x /3 for P > J-L .

116 JEAN A. LARSON

With GCH in hand, one can construct a sequence (Xa I 0: < K,) that lists allsubsets of AX ~ for all limit ordinals ~ < K, of cofinality ~. At stage [3, Hajnal alsomade sure that if ~ :s; [3 were a multiple limit point of some X1/ with 1] < [3, theneach (f..L, (3) for f..L < ~ was joined to a point of X1/'

In the construction of this paper, the role of restrictions X n (A x ~) is takenover by ~-guessing functions defined below. One property of these functions is thatof being well-tuned.

Definition 2.11. A function 9 : A~ A is well-tuned if it is strictly increasing andg(o:) > 0: for all 0: < A.

The following fact is useful in identifying well-tuned functions.

Lemma 2.12. If a function 9 : A~ A is strictly increasing and g(o:) > 0: for alllimit 0: < A, then it is well-tuned.

Proof. Use induction on [3 to prove that g([3) > [3 for all [3 < A. The basis andlimit cases are true by hypothesis. The successor cases are also straightforward,since if [3 < g([3), then [3 + 1 :s; g([3) < g([3 + 1). 0

The construction of the counter-example for K, ' Auses (~, g, ,) -harmonious sets,whose definition in turn, uses half-open ordinal intervals. Recall that [1],() is theset of all ordinals ~ with 1] :s; ~ < (. Here is a picture; the definition follows.

,[.

[ .cob)

gb)

)

! eo

•

Definition 2.13. Suppose ~ < K, =A+ is a limit ordinal of cofinality A, 9 : A~ Ais well-tuned, eo : A ~ ~ is an enumeration of ~ and Co : A ~ ~ is a strictlyincreasing function whose range is cofinal in~. The (8,g, ,)-harmonious set definedusing eo and Co is

Lemma 2.14. Suppose ~ < K, = A+ is a limit ordinal, eo : A~ 8 is an enumeration of ~ and Co : A~ ~ is a strictly increasing function whose range is cofinalin~. Then the (8, g, ,)-harmonious set defined using eo and ('5 is a subset of thehalf-open ordinal interval [co(r),~) of cardinality less than A.

Definition 2.15. Suppose K, = A+, a is a multiple limit point of X ~ A x K"

eo : A~ a is an enumeration of a and Co : A~ a is a strictly increasing functionwhose range is cofinal in a. Then g : A~ A is a a-guessing function for X usingeo and Co, if g is well-tuned and there is some Po < A called the threshold so that

1. (fiber guessing) for all TJ ~ Po, there is J.t E [TJ, g(TJ)) so that a is a limit pointof X/L ; and

2. (point guessing) for all , ~ Po, if J.t < , and a is a limit point of X/L' thenX n ({J.t} x R(a,g,,)) :f; 0 where R(a,g,,) is the (a,g,,)-harmonious setdefined using Co and eo.

Lemma 2.16. Suppose K, = A+, a< K, is a limit ordinal of cofinality A, eo: A~ ais an enumeration of a and Co : A~ a is a strictly increasing function whose rangeis cofinal in a. If a is a multiple limit point of X ~ Ax K" there is a a-guessingfunction for X using eo and co '

Proof. Use recursion on , to define the required a-guessing function. o

Functions can be compared by the relation of eventual domination.

Definition 2.17. A function g : A~ A is eventually dominated by f : A~ A,in symbols g <* I, if there is some P < A such that g(a) < f(a) for all a ~ p. Ifg <* i, then f3 < A is above the threshold for {g, f} if g(a) < f(a) for all a ~ f3 .Lemma 2.18. Suppose K, = A+, a< K, is a limit ordinal of cofinality A, eo : A~ ais an enumeration of aand Co : A~ a is a strictly increasing function whose rangeis cofinal in a. If a is a multiple limit point of X ~ Ax K" g is a a-guessing functionfor X using eo and Co, g <* h, and h is well-tuned, then h is a a-guessing functionfor X using eo and co'

Definition 2.19 (see p. 115 of [24]) . A collection V ~ .\A is called a dominating family if for every function g : A~ A there is some f E V so that f eventuallydominates g, in symbols, f <* g.

Definition 2.20 (see p. 115 of [24]). A scale in xA is a dominating family Swhich is well-ordered by <*. It is short if it has order type K, =A+. It is well-tunedif every member f E S is well-tuned.

The dominating number, (I, is the minimum size of a dominating family in "t» ,

Note that N1 ~ b ~ (I ~ c, where c =2No is the cardinality of the continuum, andb is the bounding number. There is a dominating family in W w well-ordered under<* if and only if b = (I . However, it is consistent that b < (I . See [24] for details.

The basic recursive construction of a short scale from a dominating family ofsize N1 in W w can easily be modified to construct a well-tuned short scale in .\A forany regular cardinal A.

Lemma 2.21 (see pp. 116-119 of [24]). If (I = N1, then there is a short welltuned scale in "t»,

The same argument generalizes for regular uncountable cardinals.

Lemma 2.22. If A is regular and there is a dominating family of size K, = A+ inxA, then there is a short well-tuned scale in xA.

118 JEAN A. LARSON

Definition 2.23. A ladder is on a limit ordinal 8 is a strictly increasing functionCo: cf 8 ~ 8 whose range is cofinal in 8. Given a subset S ~ K, consisting of limitordinals, a ladder system is a sequence (co I8 E S) so that Co is a ladder for each8 E S.

Eklof [8] has commented on the origin of the concept of ladders. See Devlinand Shelah [6] for a 1978 use of the terminology.

For the rest of this paper, assume A is regular and that there is a dominatingfamily in xA of size K, = A+. Let :F =(fa Ia < K,) be a fixed well-tuned scale. LetC = (co I8 < K, &cf 8 =A) be a ladder system. Furthermore, for each limit ordinal8 < K" let eo : A~ 8 be a fixed one-to-one onto function enumerating 8 in ordertype A. With these fixed functions in hand, assume that all 8-guessing functionsare defined using eo and co'

Lemma 2.24. Suppose that K, = A+ and A is regular. For any graph H = (A xK"F) , let 4'1(H) be the following statement: for every x = (JL,{3) E A x K" 8, andY ~ 8 x 8, if JL < 8 ~ (3, 8 is a multiple limit point ofY, and f{3 is a 8-speculationfunction for Y, then nhb+(x) n Y f:. 0. If H = (A x K" F) is a graph satisfying4'l(H), then H has no independent sets of order type K, ' A.

Proof. Suppose H = (V,F) is a graph on V = A x K, satisfying 4'1 (H). Furthersuppose 8 < K, is a limit ordinal of cofinality A, Y ~ A x 8, and JL < 8 are arbitraryso that 8 is a multiple limit point of Y. Use Lemma 2.16 to find a 8-guessingfunction g for Y. Let a be chosen so that g <* fa . Then for all {3 ;::: a, g <* f{3, soby Lemma 2.18, f{3 is also a 8-guessing function for Y. Thus by the hypothesis ofthe lemma, nhb+ ((JL, (3)) n Y i 0 for all JL < A and (3 ;::: a. Set b(8, JL, Y) = a forall JL < A. Since 0, Y and m were arbitrary, such a bound can always be defined.Thus 4'o(H) holds and the lemma follows from Lemma 2.9. 0

Definition 2.25. Suppose a, p are ordinals and U =a x p. A graph H = (U,F)is tuneful if (1) all edges move inward; (2) it is triangle-free; and (3) the outneighborhood of every point is fiber finite.

Theorem 2.26. If A is regular and there is a scale of length K, = A+ in xA, thenthere is a tuneful graph G = (V, E) on V =A x K, witnessing K, • A 1+ (K, • A,3)2.

Proof. Suppose that G = (V,E) is a graph which has no edges between points ofAx K, . Notice that all edges moveinward in G if and only if for each point x = (JL, (3),the out-neighborhood of x satisfies nhb+(x) ~ (A" (JL + 1)) 0 {3, where A 0 Bdenotes the pairs (a, (3) with a < (3. If all edges move inward, then by Lemma2.4, the graph is triangle-free if and only if the out-neighborhood of each pointis independent. Furthermore, the out-neighborhood of a point x is independentif and only if nhb+(x) n nhb+(y) for all y E nhb+(x). Thus to guarantee that Gis tuneful, it is enough to make sure that the out-neighborhood of every pointx = (JL , (3) for {3 < A is empty and the out-neighborhood of each x = (JL, (3) for{:J ;::: A is a fiber finite subset of (A <, (JL + 1)) 0 (:J so that nhb+(x) n nhb+(y) = 0for all y E nhb+(x) .

Define a graph G = (V,F) on V = A x K. by recursion on (3 < K.. At stage(3, define the out-neighborhood of nhb+(J.L, (3) for all J.L < A, so that the followinginduction hypotheses are satisfied:

1. nhb+(J.L, (3) ~ (A" (J.L + 1)) ® (3 with nhb+(J.L, (3) = 0 if (3 < Aj2. nhb+ (J.L, (3) has small fibers ;3. nhb+(J.L,(3) nnhb+(y) = 0 for all y E nhb+(J.L,(3) j and4. if fJ ~ (3, y ~ fJ x fJ are such that fJ is a multiple limit point of Y and 113 is a

fJ-guessing function for Y , then nhb + (J.L, (3) n Y ::j; 0.To start the recursion, let nhb + (J.L , (3) = 0for all J.L, (3 < A. Notice that trivially,

all four conditions are satisfied.To continue the recursion, suppose that (3 ~ A and for all y = (V,7) E A x (3,

nhb + (y) has been defined so that the induction hypotheses are true.Let (dp ,o I Q < A) be a sequence so that dp,o ~ (3 is a limit ordinal for

all Q < A, and each limit ordinal fJ ~ (3 is equal to dp.o for A many Q. Let

(Pp,o I Q < A) and ( P/J.o I0 < Q < A) be defined by recursion on Q as follows:

set Pp,o = OJ p~.o+l = Pp,o and PP,o+l = Ip(P/J,o) for successor ordinals; andP~,o = sUP"y<o Pp,o and Pp,o = Ip(P/J.o) for limit ordinals.

For each J.L < A, define nhb + (J.L, (3) n ({p} x (3) by recursion on p. For p ~ J.L,set nhb+ (J.L , (3) n ({p} x (3) = 0. Suppose p > J.L and nhb + (J.L, (3) n ({p'} x (3) hasbeen .defined for p' < p. Let Q > 0 be such that p~.o ~ P < Pp,o . Since 113is strictly increasing, there must be a unique Q with this property. Finally setnhb+(J.L,(3) n ({p} x (3) = {p} x R where R = R(dp,o,113,7) is the (dp.o,/p,7)harmonious set of Definition 2.15 and 7 > P is chosen so that Cdl3,a > 7 andCdl3.a ~ 17 for all z = (P,17) in ({p} x (3) n U{ nhb" (y) lyE nhb'l'{rn, (3) n (p x (3) }.By the induction hypothesis on p, nhb + (J.L, (3) n (p x (3) has small fibers, hence hascardinality less than A, so the desired 7 can always be found .

x

de,«

R(dp.o'/p, 7)

co(r)

(J.L, J.L) • • •• •• • • •.••••• •••• • •••

120 JEAN A. LARSON

By the induction hypothesis on p and the choice of R = R(d{3,oo f{3, 'Y), itfollows that for x = (IL, (3), nhb + (x) n ((p + 1) x (3) is a subset of (A <, (IL +1)) x (3 and nhb+(x) n nhb+(y) = 0 for all y E nhb+(x) . Moreover, R(d{3,a, f{3, 'Y)has cardinality less than A by the definition of (d{3 ,a,J13, 'Y)-harmonious set, sonhb+ (m, (3) n ((p + 1) x (3) also has small fibers . Suppose that Y ~ A x d13 ,a hasd13,a as a multiple limit point and as a limit point of Yp , that fl3 is a d13,a-guessingfunction for Y , and that P/J,a is above the threshold for fl3' Y . By Definition 2.15

and the choice of R = R(d13,a, fl3' 'Y), it follows that nhb+(IL,(3) n Y # 0.This completes the definition by recursion on p of nhb+ (IL, (3) and the analysis

that shows by induction that the out-neighborhood satisfies the first three induction hypotheses listed above . Since for any limit 8 ~ (3 and Y ~ A x 8, if 8 is amultiple limit point of Y and fl3 is a 8-guessing function for Y, then there is an Q

so that both 8 = d13,a and P/J,a is above the threshold for Y, fl3 ' it follows that theout-neighborhood of (IL, (3) also satisfies the last induction hypothesis listed above .

Since at each stage (3 the out-neighborhoods of (IL ,(3) for IL < A have beendefined satisfying the induction hypotheses, it follows by induction that the outneighborhoods of x for all x E A x I>, have been defined satisfying them. Thereforeby Lemmas 2.4 and 2.24, the graph G = (V,E) defined by this recursion throughthe definition of its out-neighborhoods is the desired tuneful graph witnessingI>, • A -It (I>, • A, 3)2. 0

Corollary 2.27 (Restatement of Theorem 1.4). If I>, = A+ and there is ascale of length I>, in AA, then I>, • p -It (I>,. A,3)2 for all infinite cardinals A andordinals p < 1>,.

Proof. Let p be given and let G = (W, E) be the graph of the previous theorem,where W = A x 1>,. If p < A, then H = (U,0) is a witness to the corollary forU =p x 1>,. Otherwise, define H = (U, F) by

F ={((ep(lL) , (3), (ep(v), 'Y)} I {(IL, (3), (v, 'Y)} E E} .

Since the mapping 11' that takes (IL, (3) to (ep(IL), (3) is one-to-one and onto, itfollows that H is triangle-free. Furthermore, if X ~ U has order-type I>, • A, thenY ={(IL, (3) I (e, (IL), (3) EX} also has order type I>, • A, since Amany fibers havecardinality 1>,. Since G has no independent sets of that order-type, it follows thatH does not either. Thus H proves the corollary. 0

3. The example for 1>,2

The goal of this section is to adapt the construction of the previous section to thelarger product W = I>, x I>, for I>, = A+ and use the well-tuned scale :F to producea tuneful graph witnessing 1>,2 -It (1),2,3)2 .

A set A ~ W has order type 1>,2 under the lexicographic ordering if and onlyif I>, many fibers of A have cardinality 1>,. Recall that A ~ B denotes the set of allpairs (Q,(3) E A x B with Q < (3 when A and B are non-empty. If A =0 or B =0,then let A ~ B denote the 0 for notational convenience.

The following lemma characterizes graphs on W in which all edges move inwardin terms of the behavior of their out-neighborhoods.

Lemma 3.1. For any graph H = (I\: x I\:,F), all edges of H move inward if andonly if nhb+(f..L, {3) ~ «(3 <, (f..L + 1)) @ (3 for all (f..L , (3) in W = I\: X 1\:.

Definition 3.2 (Hajnal [15]). For any limit ordinal 8 and any set X ~ I\: X 1\:,

the ordinal 8 is a double limit point of X if 8 has cofinality A and for unboundedlymany TJ < 8, 8 is a limit point of XT/ .

A proof of the next lemma is included for the sake of completeness.

Lemma 3.3 (Hajnal [15]). Suppose I\: = A+ and A is regular. For any X ~ I\: X I\:

of order type 1\:2 and any f..L < 1\:, there is some limit ordinal 8 with f..L < 8 < I\: sothat 8 is a double limit point of x.Proof. Suppose X ~ I\: X I\: has order type 1\:2 and f..L < I\: is given. Use recursionon a < A to define a sequence (OOt Ia < A) as follows. Let 00 be the least ordinalo> f..L so that X(J has cardinality 1\:. Suppose 0{3 has been defined for (3 < a so that(1) (0{3 I (3 < alpha) is strictly increasing, (2) X(J., has cardinality I\: for all 'Y < a ,

and (3) X n ({O')'} x [Op ,0{3)) i= 0 for all 'Y < (3 < a, where Op = sUPT/<{30T/'

Let OOt be the least 0 so that 0 > O~ = sUPT/<Ot 0T/' X(J has cardinality I\: andX n ({O')'} x [O~ , 0)) i= 0 for all 'Y < (3 < a. Note that this choice of OOt guaranteesthat the induction hypotheses are satisfied.

Finally, let 8 = SUPOt<>' OOt . By construction, 8 is a double limit point of X . 0

Use Lemma 3.3 and modify the proof of Lemma 2.9 to prove its analog below.

Lemma 3.4. Suppose I\: = A+ and A is regular. For any graph H = (I\: x I\: ,F) ,let'l/Jo(H) be the following statement: for every limit ordinal 8 < I\: of cofinality A,every f..L < 8, and every set Y ~ 8 x 8 for which 8 is a double limit point, there issome b(8, f..L, Y) so that nhb+«f..L, (3)) nY i= 0 for all (3 with b(8,u; Y) ~ (3 < 1\:. Forany graph H = (I\: X 1\:, F) , if'l/Jo(H), then H has no independent sets of order type1\:2.

Reshape the concept of 8-guessing function to fit the new circumstances.Definition 3.5. Suppose 8 < I\: is a limit ordinal which is a double limit point ofX ~ I\: X 1\:. Then g : A~ ,\ is a 8-speculation function for X if 9 is well-tuned,and there is some Po < ,\ called the threshold of 9 and X so that for all P ~ Poand for R := R(8, g, p) the (8,g, p)-harmonious set of Definition 2.13,

1. (fiber speculation) there is TJ E R so that 8 is a limit point of XT/; and2. (point speculation) if 8 is a limit point of XT/ for TJ = e.s(x) where X < p, then

Xn({TJ}xR)i=0.Lemma 3.6. For all limit ordinals 8 < 1\:, if 8 is a double limit point of X ~ I\: X 1\:,

there is a 8-speculation function for X.

Proof. Use recursion on p to define the required 8-speculation function . 0

Lemma 3.7. For all limit ordinals 8 < 1\:, if 8 is a double limit point of X ~ ,\ X 1\:,

9 is a 8-speculation function for X, 9 <*h, and h is well-tuned, then h is a 8speculation function for X .

122 JEAN A. LARSON

Use Lemmas 3.6,3.7, and 3.4, and modify the proof of Lemma 2.24 to proveits analog below.

Lemma 3.8. Suppose", = A+ and A is regular. For any graph H = ('" X n; F), let1/Jl(H) be the following statement: for every x = (p.,{3) E '" x ",,8, and Y S;; 8 x 8,if P. < 8 $ {3, 8 is a double limit point of Y, and f(3 is a 8-speculation function forY, then nhb+(x)nY i' 0. If H = (A x A+,F) is a graph satisfying 1/Jl(H), then Hhas no independent sets of order type A+ . A.

Lemma 3.9. Suppose", = A+ and A is regular. Further suppose that x = (p.,{3) E'" x n, and for all y = (v, ')') E '" x {3, the out-neighborhood of y has been definedso that nhb+(y) S;; (')'" (v + 1)) ® ')' and nhb+(y) has small fibers. Then nhb+(x)can be defined so that

(1). nhb+ (z)' S;; ({3 -, (p. + 1)) ® {3j(2). nhb+(x) has small fibers;(3). nhb+(x) n nhb+(y) =0 for all y E nhb+(x)j and(4). if 8 and Y S;; 8 x 8 are such that p. < 8 $ {3, 8 is a double limit point of Y,

and f(3 is a 8-speculation function for Y, then nhb+(x)n Y i' 0.

Proof. Since condition (4) is the only one which requires nhb+(x) to be non-empty,if there is no limit ordinal 8 of cofinality A with p. < 8 $ {3, then nhb+(x) := 0satisfies all the conditions.

So suppose there is such an ordinal 8. In this case, let ~ = (81/ 171 < A) be asequence so that cf 81/ = A and p. < 81/ $ {3 for all 71 < A and so that if cf 8 = Aand p. < 8 $ {3, then 8 = 81/ for A many 71 < A. Clearly ~ depends on x, but allmention of x is suppressed in favor of notational simplicity.

Use recursion on 71 < Ato define nhb't'{z}. At stage 71, define Q1/:= R(81/,f(3'')'1/)and for each 11" E Q1/' define N1I" := {11"} xR(81/' f(3,01l") by chosing-y; and 011" minimalso that the following conditions hold.

(a). 71 < ')'1/ and p. < C6.,(')'1/);(b). /(3(')'1/) < 011" and 11" < C6., (011") for all 11" E Q1/ ;(c). for all v E U{ Qa Ia < 71}, either v < C6.,(')'1/) or 81/ $ v;

(d). for all 11" E Q1/' V E U{ Qa I a $ 71}, yEN", and z = (11", () E nhb+(y), ifv < 11", then either « C6.,(01l") or 81/ $ (.

Here is a picture ofthe relationship between Ua <1/ Qa and the interval [C6., (')'1/), 81/)used in the definition of Q1/'

10 0 0 )000)

The recursion continues for all 71 < A since at each stage there are fewer thanA many conditions to be satisfied by ')'1/ and by 011" with 11" E Q1/' Finally, setnhb+(x):= U{N1I" 111" E Q1/&71 < A}.

Claim 1. nhb+(x) S;; ({3 <, (p. + 1)) ® {3.

Suppose 1] < A and 7r E Q1) are arbitrary. From (a) and Definition 2.13 itfollows that J.I. < C6,/'Y1)) ~ 7r and R(81),flJ,87f) ~ [c6.,(87f ), 81) ) ~ (3. However7r < C6., (87f) < 81) ~ {3 by condition (b), so N7f ~ ({3, (J.I. + 1)) @(3. Since 1], 7r werearbitrary, the claim follows.

Claim 2. nhb+(x) has small fibers.

If a < 1] < A, then Qcr n Q1) = 0 by condition (c). Hence for all 1] < Aand all 7r E Q1)' the 7rth fiber of nhb+(x) satisfies nhb;(x) = N7f. Since eachN7f := {7r} x R(81),flJ,87f) and R(81),flJ,87f) has cardinality less than A by Lemma2.14, it follows that nhb+(x) has small fibers.

Claim 3. nhb+(x) n nhb+(y) = 0 for all y E nhb+(x).

Suppose 1] < A, 7r E Q1) and y = (v, "y) E nhb+(x) are arbitrary. Let a < A besuch that v E Qcr and yENII' If 7r ~ v, then N7f n nhb+ (y) = 0since N7f ~ {7r} X (3and nhb+(y) ~ b' (v+l))@"Y. If v < 7r and a ~ 1], then N7f n nhb+ (y) =0by thedefinition of N7f = {7r} x R(81), flJ,87f)' Lemma 2.14 and condition (d). If v < 7r and1] < a, then by condition (c), it follows that 8cr ~ 7r, so "Y < 7r and consequentlyN7f n nhb+(y) = 0 since nhb+(y) ~ (-y, (v + 1)) @ "Y. Because either (7r ~ v) or(v < 7r and a ~ 1]) or (v < tt and 1] < a), it follows that N7f n nhb+(y) = 0.Since y was arbitrary, it is true for all y E nhb+(x) . Hence the claim follows bythe definition of nhb+(x).

Claim 4. if 8 and Y ~ 8 x 8 are such that J.I. < 8 ~ (3, 8 is a double limit point ofY , and flJ is a 8-speculation function for Y, then nhb+(x) n Y :f;0.

Suppose that 8 and Y satisfy the hypotheses of the claim. Let 1] be such that1] is above the threshold for Y and ff3 and 81) = 8. Such a choice is possible by thedefinition of ~. Since 1] < "Y1)' it is also above the threshold for Y and ff3. Thusby Definition 3.5, there is 7r E Q1) =R(81) , flJ, "Y1)) so that 81) is a limit point of Y7f.Let ~ be such that es; (~) = -s , Note that ~ < h(-yeta) by Definition 2.13. Now"Y1) < f{3(-ye ta) < 87f, since f{3 is well-tuned. Thus 87f is above the threshold for Yand I». so Y n N7f :f; 0 by Definition 3.5, which suffices for the proof of the claim.

The above four claims complete the proof of the lemma. 0

Theorem 3.10 (restatement of Theorem 1.17). If K, = A+, A is regular andthere is a short scale in AA, then there is a tuneful graph G = (V, E) on V = K, x K,

witnessing K,2 f+ (K,2, 3)2 .

Proof. As in the proof of Theorem 2.26, define a tuneful graph G = (W,E) byrecursion on {3 < K,. At stage {3, define the out-neighborhood of (J.I., (3) for allJ.I. < K" so that the four conditions of Lemma 3.9 are satisfied.

To start the recursion, set nhb+ (J.I., (3) := 0 for all {3 < A and notice that thefour conditions are trivially satisfied . For {3 ~ A, use Lemma 3.9 applied to (J.I., (3)for all J.I. < K,.

Let E = U{nhb+(J.I.,{3) I (J.I.,{3) E K, x K,} . By Lemma 3.1, all edges of E moveinward. Since all out-neighborhoods satisfy condition (3) of Lemma 3.9, they areindependent. Since all edges move inward, it follows from Lemma 2.4 that G has

124 JEAN A. LARSON

no triangles. Consequently, since all out-neighborhoods satisfy condition (2) ofLemma 3.9, it follows that G is tuneful. Furthermore, since all out-neighborhoodssatisfy condition (4) of Lemma 3.9, the graph G satisfies 'l/Jl(G), so it has noindependent subsets of order type ",2 by Lemma 3.8. Therefore G witnesses thatthe theorem is true . 0

4. Final remarks

If CR holds then Wl kl f+ (Wl k+l, l + 1)2 for all positive integers k and l. Onthe other hand, just using the regularity of Wl, one can show that Wl kl+1 -7

(Wl k+l, l + 1)2 for all positive integers k and e.See [9] or [10J.The proofs of the negative partitions use the fact that with CR one can get a

tuneful graph on Wl x Wl so that for every fL < v < {3 with {3 infinite, there is onlyone edge from (fL, (3) ending in a point of {v} X Wl.

It would be interesting to know if the Souslin tree construction or the construction from a scale or from a morass have similar consequences for finite powers ofcardinals.

References

1. J.E. Baumgartner, Improvement of a partition theorem of Erdos and Hajnal, Journal of Combinatorial Theory (A) 17 (1974),134-137.

2. J.E. Baumgartner, Partition relations for uncountable ordinals, Israel J. Math. 21(1975) 296-307 .

3. J.E. Baumgartner, Remarks on partition relations for Wl, in: Set Theory and itsapplications (Toronto, ON, 1987), J. Steprans and W. Stephen Watson eds., LectureNotes in Math. 1401, Springer (1989).

4. J.E. Baumgartner and A. Hajnal, A remark on partition relations for infinite ordinals , with an application to finite combinatorics, in: Logic and Combinatorics,S.G. Simpson, ed., (Proceedings of a Summer Research Conf., Arcata, Ca. 1985),Contemporary Math. 65 (1987) 157-167.

5. K. Devlin , and H. Johnsbraten, The Souslin problem, Lecture Notes in Mathematics405 Springer-Verlag, Berlin-New York, 1974.

6. K. Devlin, and S. Shelah, A weak version of 0 which follows from 2No < 2Nl , IsraelJ. of Mathematics 29 (1978) 239-247.

7. B. Dushnik and E.W. Miller, Partially ordered sets , American J. of Math. 63 (1941)605.

8. Eklof, Paul C., Set theory generated by Abelian group theory, B. Symbolic Logic 3no. 1 (1997) 1-16 .

9. P. Erdos and A. Hajnal, Ordinary partition relations for ordinal numbers, PeriodicaMath . Hung. 1 (1971) 171-185.

10. P. Erdos, A. Hajnal and J . Larson, Ordinal Partition Behavior of Finite Powers ofCardinals, in: Finite and Infinite Combinatorics in Sets and Logic, N.W. Sauer, R.E.Woodrow and B. Sands, eds. , (Proceedings of the NATO Advanced Study Institute,Banff, Canada, 1991), NATO ASI Series 411 (1993) 97-115.

11. P. Erdos and R. Rado, Combinatorial theorems on classification of subsets of a givenset, Proc. London Math. Soc. (3) 2 (1952) 417-439.

12. P. Erdos and R. Rado, A partition calculus in set theory, Bull . Amer. Math. Soc.62 427-489 (1956).


13. P. Erdos and R. Rado, Partition relations and transitivity domains of binary relations, J. London Math. Soc. 42 624-633 (1967).

14. L. Haddad and G. Sabbagh , Nouveaux resultats sur les nombre de Ramsey generalises C.R . Acad. Sci. Paris Sir. A-B 268 (1969) A1516-A1518.

15. A. Hajnal , A negative partition relation, Proc. Nat. Acad. Sci. USA 68 (1971) 142144.

16. R. Jensen, The fine structure of the constructible universe, Ann. Math. Logic 4(1972) 229-308.

17. J . Larson, Partition theorem for certain ordinal products, in: Infinite and FiniteSets, (Keszthely, Hungary) Colloquia Mathematica Socitatis Janos Bolyai 10 (1973)1017-1024.

18. J .A. Larson and W.J. Mitchell, On a problem of Erdos and Rado, Annals of Combinatorics, to appear.

19. T . Miyamoto, Some results in forcing, Doctoral Dissertation, Dartmouth College,1987.

20. S. Shelah and L. Stanley, A theorem and some consistency results in partitioncalculus Ann. Pure Appl. Logic 36 (1987) 119-152.

21. S. Shelah and L. Stanley, More consistency results in partition calculus, Israel J.Math. 81 (1993) 97-110.

22. E. Specker, Teilmengen von Mengen mit Relationen, Comment. Math. Helo. 31(1957) 302-314.

23. L. Stanley and D. Velleman; Charles Morgan, W3 Wl -t (W3 Wl, 3)2 requires an inaccessible Proc. Amer. Math. Soc. 111 (1991) 1105-1118.

24. E.K. van Douwen, The integers and topology, in: Handbook of Set Theoretic Topology, 2nd printing, K. Kunen and J .E. Vaughan eds., North-Holland and ElsevierScience Publishers, New York, NY (1988), 111-167.

A PICARESQUE APPROACH TO SET THEORY GENEALOGY

JEAN A. LARSON! (September 18, 1997)Department of MathematicsUniversity of FloridaGainesville, FL 32611USA

Abstract. The relation between thesis advisor and doctoral student can be usedto draw up academic genealogies. This article discusses brief genealogies for manyof the participants (mainly set theorists) of the conferences in Curacao in 1995and Barcelona in 1996 and of the contributors to this volume. It includes a partiallist of students of A. Mostowski, some stories of mathematical beginnings andreferences for those interested in further study.

1. Introduction

Someone, likely John Addison", told me when I was a graduate student that asignificant percentage of the people in logic, set theory and foundations in theUnited States descended from either Alonzo Church or from Alfred Tarski. WhenI started drawing up lists of participants for the 1995 conference in Curacao, Iremembered this idea and decided to try and develop some genealogies for thepeople who participated in the conference. I started with the participants andlooked back to their advisors and the advisors of advisors, constructing trees todisplay the information. Since then, I have enriched the sample, including theparticipants of the 1996 Barcelona conference. To simplify construction of thetrees, I looked for the designated advisor of record, or the person mentioned inobituaries and historical articles. In cases of co-advisors, I placed the individualin a tree with the senior advisor and credited the other co-advisor in the text. Ihave used bold type to highlight the names of advisors, formal or not, who do not

lThis work was partly supported by grants number DMS-9306286 and INT-9503676from the National Science Foundation.

21 would like to thank Addison for his role as an informal advisor when 1 was anundergraduate at the University of California in Berkeley. Partly on his recommendation1 took classes from Stephen Cook, Leon Henkin and Robert Vaught; 1applied to graduateschool only after Addison suggested 1 do so.

127

C.A. Di Prisco et ai. (ed.), Set Theory, 127-155.© 1998 Kluwer Academic Publishers.

128 JEAN A. LARSON

appear in a displayed tree. In the trees, participants names are superscripted withB for Barcelona and C for Curacao.

Here are the results of this enjoyable effort. Section 2 focuses on direct andcollateral relatives of Church; Section 3 gives ancestors and descendents of Tarski;Section 4 gives ancestors and descendents of Hilbert, including Solovay and hisdescendents. Section 5 discusses those whose trees did not hook up with any ofChurch, Tarski or Hilbert. The final section looks back at the results.

2. Church

The first advisor tree , Figure 1, givessome of the descendents of Alonzo Church.A total of thirty-one individuals earned Ph.D. 's with him, most at Princeton and afew at UCLA. He was one of the principal founders of the Association of SymbolicLogic and served as editor for reviews of the first forty-four volumes of the Journalof Symbolic Logic. See H. Enderton [41] for details, including a list of the doctoralstudents of Church. W. Hodges [60] and H. Barendregt [12] have articles in Volume3 of the Bulletin of Symbolic Logic on the mathematics of Church, and a futureissue will have articles about by C.A. Anderson on philosophy of Church and byH. Enderton on Church's role at the Journal of Symbolic Logic and in early daysof the Association for Symbolic Logic.

In a 1974 conversation on the rise of mathematical logic (see Crossley [33]),Stephen C. Kleene (see Figure 1) said that he learned logic by learning a system ofChurch that was later shown inconsistent, and that he only began to read much ofthe literature after his degree. J. Schoenfeld [111] discusses Kleene's mathematicalwork and mentions some non-mathematical events in his life including a logictreasure hunt organized by John Addison in 1956. A biographical sketch of Kleene,his bibliography up to 1980 and a list of his students appears in [14]. S. Mac Lane[79] has a charming reminisce with quotes from early letters of Kleene. In [72],Kleene (see Figure 1) discussed the impact of Godel's work on graduate studentsat Princeton in the 1930's.

Michael O. Rabin (see Figure 1) earned a master's degree at Hebrew Universityin 1953 before his doctoral work in Princeton.

Dana S. Scott (see Figure 1) dated his interest in mathematics to the end ofjunior high and beginning of high school in Chico, California. His band teacher,LeslieSweeney, gave him a book on musical acoustics, and to understand the book,he had to teach himself more mathematics.

Dana Scott was introduced to logic as an undergraduate at the University ofCalifornia, Berkeley:

I had a job working in the Periodicals Room of the Main Library. At thattime there were not so many scholarly journals, and they did not have to bestored in specialized annexes. While shelving issues one day I came across avery strange title: The Journal of Symbolic Logic. I checked out some copiesand found most of the articles far beyond what I could understand. However,there was one paper by one Jan Kalicki, who was, I realized, going to teach

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a course the next semester at Berkeley. I signed up for that course, and theinstructor was very startled to have a sophomore come up to him and say hehad read one of his papers. This was the beginning of a close friendship, andProfessor Kalicki not only introduced me to the idea of research, but he alsointroduced me to . .. Alfred Tarski, and his very active circle of students. Inthis way I came to find out about the most recent research in mathematicallogic at the time, and the strong influence of Tarski set me on my own coursein research. [112]

When I asked John W. Addison how he got into the field, he started his replyby mentioning that he took as a junior a logic course from Church who usedIntroduction to logic and to the methodology of deductive sciences by Tarski [117]and he thought that Martin Davis, who earned his Ph.D. with Church in 1959,was the reader for the course. He continued:

I liked the book and the material, did very well in the course (unlike most ofmy earlier work in mathematics at Princeton, to which I had devoted shockingly little time and effort, particularly in contrast to the time I spent on suchextracurricular activities as the Daily Princetonian, the Debate Panel, tennis,helping to manage the football team, World Federalism (to which I had been recruited by John G. Kemeny" after he in turn was recruited by Albert Einstein). .. I liked Church very much as a teacher, although I had not been inspired byhis course in projective geometry, which I had taken as a sophomore. (The textwas by Oswald Veblen, who was then at the Institute for Advanced Study, andJ .W. Young'' .. . ).After taking Church's undergraduate course in logic I decided to take duringmy Senior year his two-semester graduate sequence in logic ... and ultimately towrite my Senior Thesis (on Turing Machines) under his supervision. Church hadearlier suggested a problem in propositional logic ... on the ternary connectiveknown as "conditioned disjunction" '" as a thesis problem, but it must nothave interested me very much since I had done essentially no work on it when,about two weeks before the deadline (the thesis was supposed to have been anall-year project), William Boones, then a doctoral student of Church, came tomy rescue and suggested that I study Turing's original paper on machines.Boone also interested me that year in the isomorphism problem for cancellationsemigroups, which I found to be undecidable during my first year of graduatestudy at Wisconsin and which became the subject matter of Part One of mytwo-part doctoral dissertation under Kleene.As I think back on those times I am struck by how much interaction therewas between the undergraduates and the graduate students. In addition toDavis and Boone, another student who was very friendly and helpful to me

3Kemeny (Ph.D. with Church in 1949), as president of Dartmouth, was the one toput the hood on me when I earned my Ph .D. in 1972.

4John Wesley Young earned his Ph.D . in 1904 from Cornell University; the JohnWesley Young instructorships at Dartmouth College are named for him.

5Boone earned his degree in 1952.

PICARESQUE GENEALOGY 131

was Hartley Rogers", who was a fellow student with me in Church's graduatecourse. [3]John Steel [113] said that he started with Stephen G. Simpson? and that

Robert Solovay (see Figure 10) was also an early influence.Carlos Uzcategui (see Figure 1) earned his master's degree with Di Prisco in

Venezuela before continuing for the doctorate in California.

Since Alonzo Church was a student of O. Veblen" the advisor tree of Figure 2includes ancestors of Church, whose descendents appeared in Figure 1. Also, R.L.Moore appears in the tree, since he was the advisor of Mary Ellen Rudin, whosedescendents appear in Figure 3; similarly, G.D. Birkhoff appears in the tree, sincehis descendents include M. Victoria Marshall (see Figures 4 and 5).

Oswald Veblen

1903, U Chicago

H.A. Newton - E.H. Moore1885, Yale

Figure 2.

R.L. Moore

1905, U Chicago

G.D. Birkhoff

1907, U Chicago

While Hubert Anson Newton (see Figure 2) continued studying mathematicspast his A.B. in 1950 at Yale, both there and for one year in Europe, his obituary[97] mentions no higher degree other than an honorary one. He was known for hiswork on meteors, and was one of the original charter members of the NationalAcademy of Sciences, founded in 1863.

K. Parshall [96] has written about E. Hastings Moore (see Figure 2) and his roledirecting graduate students at the University of Chicago, including G.D. Birkhoff,L.E. Dickson, R.L. Moore and O. Veblen. A follower of John Dewey, E.H. Mooreused a laboratory style of teaching rather than the lecture method.

Aspray [7] has written on the influence of Oswald Veblen on Church, whomVeblen singled out as an undergraduate.

Traylor [121] has written about Robert Lee Moore (see Figure 2), and hasa section on his graduate student days, that discusses both E.H. Moore and O.Veblen, whose own thesis work provided the question on which R.L. Moore worked.Traylor lists the students of R.L. Moore and some of their descendents. Wilder

6Rogers earned his Ph .D. in 1953 with Church.7Simpson earned his Ph .D. at MIT with Gerald Sacks, whose Cornell Ph.D . waswith

J . Barclay Rosser (a student of Church) .8See Montegomery [88] where R.L. Moore is also called a student of Veblen

132 JEAN A. LARSON

[129], [130] declared the question of whether Veblen or E.H. Moore is the "official"advisor of R.L. Moore cannot be resolved due to lack of records. They clearlyworked together after R.L. Moore arrived in 1903. Moore got started on researchwith George Bruce Halsted at the University of Texas, where Moore earned bothB.A. and M.A. degrees in 1901.

Mary Ellen Rudin was a student of R.L. Moore, so the tree of Figure 3 connectsto that of Figure 2. See Tall [119] for a lists of her publications and students, andStarbird [122] for an article on her role as an advisor . In an interview with Albersand Reid [6], [5], Rudin said she met R.L. Moore her very first day at the Universityof Texas and took a class from him every semester of both her undergraduate andgraduate school career. She also spoke of her own teaching style:

I bubble, and I get students enthusiastic. I'm able to explain things. I'm a goodlecturer, I think. I've never tried to use the Moore method'[-I guess because Igot burned by having been a student under it (p. 132 of [6], p. 297 of [5]) .

_ Mary Ellen Rudin

1949, UT AustinFranklin D. Tall

1969, UW Madison

Figure 3.

William A.R. Weiss1975, U Toronto

Maxim R. Burke'?1988, U Toronto

Piotr Koszmider'"1992, U Toronto

Franklin D. Tall and William A.R. Weiss were joint advisors of Piotr Koszmider(see Figure 3). Winfried Just (see Figure 6) was the guide for Koszmider's master'sdegree at Warsaw University, and Koszmider [73] has dated his set-theoretic originto Just.

Next we turn to some descendents of George David Birkhoff l? who appears inFigures 2 and 4. He was sometimes alluded to as the father of American mathematics for his advocacy of it .

_ G .D. Birkhoff

1907, U ChicagoJ .L. Walsh1920, Harvard

Figure 4.

J.L. Doob1932, Harvard

9The Moore method has continued to influenceteaching: in the 1980's, James M. Henle(see Figure 11) used a modified Moore method to teach set theory to undergraduates,and wrote a textbook [59] designed to enable others to teach problem-oriented courses.

lOGarret Birkhoff was the son of G.D. Birkhoff.


_ D.H. Blackwell

1941, UI Urbana R. Chuaqui

1965, UC Berkeley

Figure 5.

M.V. Marshallf

1983, PUC Chile

The advisor of David H. Blackwell (see Figure 5) was Joseph L. Doob of Figure 4. While Blackwell is especially known for his work in game theory, Bayesianinference and information theory, he is now an emeritus professor in the Groupin Logic and the Methodology of Science at the University of California, Berkeley. He and Jan Mycielski are two pioneers in connecting game theory and settheory. Mycielski and Steinhaus proposed the Axiom of Determinacyl! (AD) asan alternative to the Axiom of Choice in a paper published in 1962 [92J. In 1967,Blackwell [17] discussed infinite Go-games; and in 1969, in a Jubilee volume forHugo Steinhaus, Blackwell [18] introduced a class of games of infinite length andimperfect information, and proved a determinacy theorem for a subclass. In 1981,Blackwell [19] gave a direct description of Borel sets based on games.

Blackwell told Albers [4] that his high school geometry teacher got him reallyinterested in mathematics. When Blackwell started college, he expected to becomean elementary school teacher, but kept postponing the necessary education courses;he earned a master's degree, and then, with a fellowship, continued for a Ph.D.See his conversation with DeGroot [37J for details.

Rolando Chuaqui (see Figure 5) studied medicine in Universidad de Chile, andobtained the degree of Medico Cirujano in 1960. Preocupied by the random character of the process of medical diagnosis, and looking for a more logic formulationof the reasoning used in this process, he traveled to the United States to study,and became interested in logic and mathematics. Tarski was an important earlyinfluence. Chuaqui was instrumental in the development of a graduate program inmathematics at Pontificia Universidad Cat6lica de Chile (PUC). For more information, see [24J and [26] , which also includes a list of his students.

3. Tarski

S. Givant [51J has written a delightful profile of Alfred Tarski (see Figures 6 and7) that includes pictures of him and some of his students. W. Hodges [64] gives a

HIn a paper partially written at Berkeley during 1961/62 and published in 1964, Mycielski [93] expressed his hope that the consistency of AD would be proved via "submodelsof the natural models" [the natural models of Montague and Vaught [87] are models defined by recursion over ordinals from a seed by taking power set at successor stages andcollecting at limit stages]. In a footnote, Mycielski mentioned that it would be "still morepleasant if such a submodel contains all the real numbers of the natural model." Mycielski's hopes were realized by Martin (see Figure 11) and Steel (see Figure 1) when theyproved that AD is true in L[R], using work of Woodin (see Figure 10). Mycielski hasfurther historical remarks in [94].

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list of doctoral students and other students on whose dissertations Tarski had asignificant influence, including the three appearing in Figure 6: Andrzej Mostowski,Robert L. Vaught and J . Donald Monk. For an earlier list of students of Tarski,with information on where they were at the time of listing, see [118]. Hodges alsolists individuals who worked with Tarski on grants from the U.S. National ScienceFoundation and other bodies. R. Chuaqui [25] and J . Addison [2] have written onTarski as well; and there are a series of articles in volumes 51 and 53 of the Journalof Symbolic Logic on various aspects of Tarski's work.

H. Rasiowa [101] reported that in the years 1936 and 1937, Andrzej Mostowskivisited University of Vienna, where he was a student of Godel and Federal Polytechnic University in Zurich where he attended lectures of Hermann Weyl (see Figure 8) on symmetry. Rasiowa called Tarski the "strongest influence" on Mostowski;and Kuratowski [75] (who may have been Mostowski's advisor of record) saidMostowski's thesis was written "under the supervision of Alfred Tarski." He alsomentions that Mostowski worked as an accountant in a tile factory during theNazi occupation of Poland. In [33], Mostowski said he was a student of Tarski anddescribed some of the mathematics he lost when he chose bread over set theorynotes.

Wiktor (Victor) Marek, who has compiled a bibliography [81] of Mostowski(see Figure 6), sent me a partial list of Mostowski's students. Since I am unawareof a list this extensive in the English language literature, I am including it here:

Roman Sikorski, Helena Rasiowa, Andrzej Grzegorczyk, Andrzej Janiczak,Andrzej Ehrenfeucht, Victor Marek, Maciej Brynski, Janusz Onyszkiewicz,Kazimierz Wisniewski, Wojciech Guzicki, Pawel Zbierski, Michal Krynicki,Krzysztof Apt, Stanislaw Krajewski, Zofia Adamowicz, Malgorzata Lachlan.

Mostowski died before the thesis of M. Lachlan was written, and she finishedher degree with W. Marek. [82]

Rasiowa [101] stated that she and Grzegorczyk were Mostowski's first students,earning doctorates in 1950; she also mentioned Henry Hiz12, Antoni Janiczak (whodied in 1953) and Andrzej Ehrenfeucht as early students of Mostowski.

During World War II Robert L. Vaught (see Figure 6) was an undergraduatestudent at Pomona and had been there for perhaps a little less than two years whenhis education there was interrupted by either a draft call or a decision to enlist. Hejoined the Navy and applied for their V-12 educational program. He was acceptedfor it and was sent to Berkeley, where he completed 3 semesters of acceleratedstudy. His major was Physics, but he did take several mathematics courses whileat Berkeley, including one by Alfred Foster!". This among other things may havekindled some interest in logic. After his V-12 studies at Berkeley were completed, heserved on a destroyer in the Pacific and while on board read Tarski's undergraduatetext Introduction to logic and to the methodology of deductive sciences 14 [117]. After

12Wolenski (131][P. 309] says that Hiz completed his studies under Warsaw logicians,particularly Lesniewski .

13Church's first doctoral student.14Vaught (125] mentions reading this book in his overview of Tarski's work in model

theory.

136 JEAN A. LARSON

he got out of the Navy, he returned to Berkeley for graduate studies. (This briefdescription is based on a telephone conversation by Addison [3] with Vaught.)

In his article on Tarski, Givant [51] described Vaught's request for a problemworthy of a thesis and the still unsolved problem that was Tarski's response.

William J . Mitchell [86] of Figure 6 has dated his interest in logic to a lectureat the University of Wisconsin by a fellow undergraduate student in the seniorseminar on GOdel's proof of the Completeness Theorem.

The Monk to Darby line exhibits an interesting alternation between Berkeleyand Boulder.

Don L. Pigozzi (see Figure 6) and Josep Marla Font were co-advisors of RaimonElgueta, who wrote a thesis in algebraic logic.

J . Lukasiewicz

1902, U Lwow

_ K. Twardowski

1892, U ViennaS. Lesniewski

1912, U Lwow

T. Kotarbhiski

1912, U Lwow

Figure 7.

A. Tarski

1924, Warsaw U

Kazimierz Twardowski (see Figure 7) studied philosophy at Vienna University in the main with Franz Brentano. His official supervisor was, however , R.Zimmermann, since Brentano had lost his chair and was only a Privatdozentafter marrying as a former Roman Catholic priest (see p. 3 and 309 of [131]).Twardowski also studied in Leipzig and Munich. He was appointed professor ofphilosophy at the University of Lwow in 1895, where he founded what Wolenski[131] called Lvov-Warsaw School in philosophy and logic. (The city of Lwow isnow L'viv in Western Ukraine, and also called Lvov and Lemberg.) Twardowskilectured on mathematical logic as early as 1898 (see p. 82 of [131] where Wolenskidescribes the lectures as the first academic level contact for Poles with the subject). According to Wolenski [131, p. 13], Twardowski was never very enthusiasticabout the subject. Twardowski set up excellent working conditions for students:he organized a seminar and set up a reading room with a large library (he donatedhis private library to the university) to which students in the seminar had keys,and he had frequent personal contacts with the members of the seminar.

Jan Lukasiewicz (see Figure 7) was one of the first two students of Twardowski.He gave lectures on the algebra of relations starting in 1907/8 (Wolenski [131, p.7] said they were the first "specialist lectures on mathematical logic" in Poland) .

Stanislaw Lesniewski (see Figure 7) also studied under Cornelius in Munich andwas influenced by Leon Petrazycki, a Polish theorist of law, according to Wolenski[131]. While a student in philosophy, Lesniewski was also studying mathematicswith J. Puzyna and W. Sierpiriski. He read the logical appendix to a work of


Lukasiewicz in 1911 and started a shift to mathematical logic. For more biographical information, see Surma [123J, who also discussed the influence of the work.

Tadeusz Kotarbiriski (see Figure 7) wrote his thesis on ethics in Mill andSpencer.

W . Hodges [64J indicates that Lesniewski, Lukasiewicz and Kotarbiriski wereall influences on Tarski. (Lesniewski, Lukasiewicz held chairs in mathematics atWarsaw University and Kotarbiriski held a chair in philosophy.) According to S.Givant [51J , Tarski was an assistant for Lukasiewicz, who is known as the father ofmultiple-valued logic. Rickey [105J notes that Tarski was the only doctoral studentof Lesniewski.

Tarski studied in Poland during a period in which logic and set theory flourished . Z. Janiszewski was a prime mover behind the establishment of a Polishschool of mathematics in Warsaw and founded the journal Fundamenta Mathematicae (first issue appeared in 1920). After the death of Janiszewski in 1920, hiscolleagues S. Mazurkiewicz and W. Sierpiriski took over the post of chief editor;Lesniewski and Lukasiewicz were on the editorial board. Hugo Steinhaus'P andStefan Banach (doctoral thesis submitted 1920to University of Lwow) establisheda center of Polish mathematics in Lwow and started the journal Stu.dia Mathematica in 1929. For more information, see A half-centu.ry of polish mathematics, byKuratowski [75J, which also includes profiles of Banach and Steinhaus, and Logicand Philosophy in the Lvov- Warsaw School, by Wolenski [131J . In addition, seearticles by Ulam!", Kac and Zygmund in The Scottish Book: Mathematics fromthe Scottish Cafe as edited by R.D . Mauldin [85J , by Ciesielski and Pogoda in [27Jand Steinhaus in [114]).

4. Hilbert

Constance Reid [103J has written a wonderful book on Hilbert which includes anappreciation of Hilbert's work by Hermann Weyl. In her preface, Reid lists someindividuals who took their doctoral degrees with Hilbert, including those thatappear in Figure 8.

The two chapters, Friends and Teachers and Doctor of Philosophy, give a picture of student life of the time and the mathematical setting in which Hilbertstarted. For example, in addition to an oral examination, there was what Reidcalls a "public promotion exercise" in which he had to defend two theses of hisown choice (in a mock battle) against two fellow students appointed to be his"opponents."

Reid records Ferdinand Lindemann as the Doctor-Vater of Hilbert. Lindemann, who proved the transcendence of 71", gave Hilbert his dissertation problem.

l11Steinhaus earned his doctorate with Hilbert in 1911.16Stan Ulam [124} earned his doctorate in 1937 at the Polytechnic Institute in Lw6w

(now State University "L'vivska polytechnica"), where his sponsors were W. Stozek andK. Kuratowski. Ulam was a graduate research professor at the Universityof Florida whenI first arrived as an assistant professor.

138 JEAN A. LARSON

Reid calls Adolf Hurwitz, whom Lindemann brought to Gottlngen, "Hilbert'sreal teacher." (Hurwitz earned his doctorate in 1880 with Felix Klein in Leipzig.)

The ancestry of David Hilbert stretches back via Ferdinand Lindemann to Felix Klein of the Erlangen program and Klein bottle fame and Julius PlUcker,after whom Plucker coordinates are named. (See the Dictionary of Scientific Bibliography [50] for details.)

B.L. van der Waerden [126] wrote of the difference the mathematical readingroom, established by Klein, made to students. In 1924, when van der Waerdenarrived at Gottfngen, such rooms were unusual. Ordinarily one filled out a formrequesting a book at the University Library and had to wait an hour or more forit to arrive.

In the tree (see Figure 8) of selected first and second generation descendents ofHilbert, individuals were included either because some descendent attended one ofthe conferences or because they appear in the commentary elsewhere in the article(Hugo Steinhaus appeared in the sections on Church and Tarski; Kurt Schiitteappears below in the discussion of ancestors and descendents of Ronald Jensen).Saunders Mac Lane is included as the advisor of Robert Solovay (see Figure 10),and Solovay is represented at both conferences by descendents. Helmuth Kneser isincluded as the advisor of Reinhold Baer (see Figure 9), whose descendents includeJorge Martinez, who attended the meeting in Curacao as director of the CaribbeanMathematics Foundation.

_ D. Hilbert

1885, U Konigsberg

H. Weyl

1908, U Gottingen

H. Steinhaus1911, U Gottingen

H. Kneser1921, U Gottingen

K . Schiitte

1934, U Gottingen

Figure 8.

S. Mac Lane

1934, U Gottingen

R. Baer

1925, U Gottingen

Hermann Weyl (see Figure 8) decided go to Gottingen at eighteen because thedirector of his high school had given Weyl [127] a letter of recommendation tothe director's cousin , Hilbert. Weyl [127] also commented on his happy summervacation studying Hilbert's Zahlbericht at the end of the first year at Gottingen.

Saunders Mac Lane (see Figure 8) started working with Paul Bernays (see G.Miiller [91]), an assistant of David Hilbert, and finished with Hermann Weyl. In [78]and [5] , Mac Lane recalls his graduate student days . He had a "wonderful" teacher


his freshman mathematics course in college: Lester Hill, who was an instructor atthe time, working for his Ph .D. At Hill's suggestion, Mac Lane took the BargePrize exam, won, and decided that "maybe mathematics was a better field thanchemistry."

Helmuth Kneser (see Figure 8) wrote a paper [74] seeking to improve the connection between Zermelo's Axiom of Choice and Zorn's Lemma. (For a scholarlydiscussion of Zermelo's Theorem, see Kanamori [69].)

_ R. Baer

1925, U GottingenP. Conrad

1951, VI Urbana-Champaign

Figure 9.

J. Marthies?

1969, Tulane U

Reinhold Baer (see Figures 8 and 9) was at University of Freiburg in the 1930's,and knew Zermelo and Husserl, the philosopher . Baer became one of the fifteenfounding members of the Gessellschaft fiir Mathematische Forschung that becamethe legal basis for the Mathematical Institute at Oberwolfach after the death of itsfounder SUs. Gruenberg [52] divides his analysis of the work of Baer into sections ontopology, Abelian groups, geometry (including foundations of projective geometryand the use of groups of motions to study geometries), and other group theory. Baer[10] also has papers in set theory, including one, Zur Axiomatik der Kaminalzahlarithmetik, dedicated to Felix Hausdorff on the occasion of his sixtieth birthday.Baer had twenty doctoral students at Urbana-Champaign and nearly thirty morein Frankfurt. In the profile by Gruenberg [52], one can read about the interactionsof Baer with many individuals cited in this note.

_ R. Solovay

1964, U Chicago

K. McAloon

1966, UC Berkeley

J. Roitman'I

1974, UC Berkeley

W.H. Woodinc

1984, UC Berkeley

Figure 10.

P. Dehornoy"

1978, U Paris

Robert Solovay was one of Mac Lane's doctoral students, so the tree of Figure 10 connects to that of Figure 8.

Judith Roitman (see Figure 10) also spent time at the University of Wisconsinworking with Mary Ellen Rudin (see Figure 3) and Ken Kunen (see Figure 1).Roitman told me a bit about how she got started in mathematics.

140 JEAN A. LARSON

When I was in junior high I read 1 2 3 . . . Infinity17 and was very struck bythe (a) fact , and (b) proof that the reals were uncountable. I thought this wasabsolutely one of the most amazing things I had ever learned. Other than thatI was not particularly interested in mathematics. But much later when I gotinterested in math I headed straight to set theory and, as soon as I learnedabout them, its applications.Kunen was visiting Berkeley for a year while Fleissner and I were graduatestudents. That was the year that Comfort's survey article'" on cardinal invariants came out. We both became very interested in set-theoretical topology, andsince none of the faculty at Berkeley were particularly interested, the sensiblething was to follow Ken to Madison and work with him and Mary Ellen.. .. Ihad only a one semester's research assistantship, so I only went for summerand the fall semester. Aki Kanamori, who had met Kunen previously at Cambridge, came to Madison too to work with Ken ... as you know, he is Mathias'student.It was in Madison that I met Mary Ellen, who of course was a seminal influence,both mathematically and personally, on many mathematicians.. . . Those wereexciting times. There were a large number of students at Madison workingwith Mary Ellen and Ken. As we grew up we would come back for visits inthe summer, along with other mathematicians who got drawn into her orbitafter graduate school - Eric van Douwen was the most notable of these, butthere were many others. One year there were so many of us that Mary Ellenarranged apartments for us and we had a seminar (I think every day, surelymore than twice a week) which Mary Ellen would begin by saying "who proveda theorem since last time?" and often more than one hand was raised. [109]

Two books that Hugh Woodin saw early that sparked his interest in set theorywere 1,2,3 Infinity (mentioned by Roitman) and Cohen's [28] book on the independence of CH. He had to order Cohen 's book at the local bookstore much totheir amusement (see [132]).

5. More genealogies

This section is organized by geography in part. It starts with descendents of D.A.Martin (USA). It moves south to a tree of two Brazilians, Edison Farah and OfeliaT . Alas. Continuing the Latin theme, it includes trees with roots in Spain (Dou andMosterin) and those with roots in France (Denjoy and Hadamard). The lineage ofJoan Bagaria of Barcelona traces through Israeli mathematicians to A.A. Fraenkel.Next are some trees with roots in Hungary: the lineage of Blass traces back to F.Riesz and that of Komjath traces back to L. Kalmar, and possibly to Fejer . Thesection concludes with mixed British and German trees: ancestors and descendentsof Ronald Jensen and the Schur-Rado-Milner line.

17Gamov [49].18See Comfort [31].


D.A. Martin - E. KleinbergRockefeller

1969

c. Di priSCOBq'"MIT

1976

J. Henle?

MIT

1976

J. Llopis?

UC Venezuela

1989

G. Mendezf

UC Venezuela

1990

Figure 11.

D.A. Martin (see Figure 11), advisor of Eugene Kleinberg, never earned adoctorate, but did spend time at Harvard as a junior fellow. Richard Guy, similarlywithout a doctoral degree, called himself a "mathematical orphan."

Carlos Di Prisco (see Figure 11) described his experience as a graduate studentat MIT in the 1970's:

I got interested in logic and set theory when I was beginning graduate school atMIT. I took a set theory course taught by Hartley Rogers, which started withsome naive set theory and continued through the consistency and independence of CH. The text book was Cohen's book Set Theory and the ContinuumHypothesis [28]. Rogers' teaching impressed me very much. His lectures werevery well prepared and delivered with force and enthusiasm. William Zwickerwas the grader for that course. I liked the material very much, so I decided totake more logic courses. I took a series of courses from Kleinberg, including aModel Theory course in which he presented Morley's theorem on categoricitywhich had been recently proved. The grader for this course was Leo Harrington. I also took courses from Gerald Sacks, Greg Cherlin and Alekos Kechris.George Metakides, Anne Leggett, Gershon Sageev, Philip Lavori and DavidPincus were also at MIT during that period. At the time there were manygraduate students interested in logic at MIT; the logic seminar gathered 15to 20 participants, some of them coming from other universities in the area.Among the advanced graduate students were Harrington, Richard Shore andDave McQueen, other graduate students in logic were Fred Abramson, JamesHenle, Everett Bull, Zwicker, Sy Friedman, and a little later, Arthur Apter ,Mitchell Spector and Dave Dorer [39].

Edison Farah

1950, U Sao PauloOfelia T. Alas?

1968, U Sao Paulo

Figure 12.

142 JEAN A. LARSON

Alas gave me information on her own degree and her advisor . It would beinteresting to know the origin of the Sao Paulo group in set theory and topology.

Alberto Dou - Juan Carlos Martlnea''1983, U Complutense de Madrid

Figure 13.

Alberto Dou (see Figure 13) was educated in Madrid and Barcelona and earnedboth a Dr.Eng . in civil engineering and a Ph.D. in mathematics [61]1 In a Festschiftfor Dou based on a meeting in Madrid in 1988, Miguel de Guzman [53] discussed thework of Dou in such areas as artificial intelligence, truth, the evolution of scientificthought, and the impact of science and technology on society; and J.J. Diaz [40]discussed Dou's work in various areas of applied mathematics. J.C. Martinez (seeFigure 13) was formally a student of Dou and worked informally with Jorg Flum,of Freiburg Universitat.

Jesus Mosterin - Ignasi JaneB

1982, U Barcelona

Figure 14.

Ignasi Jane (see Figure 14) spent five years in Berkeley working with JackSilver (see Figure 6) before finishing a degree in Barcelona.

Next are trees whose roots are in France .

A. Denjoy

ENS Paris

1909

E. Corominas

Paris

1952

M. Pouzet

CBU Lyon

1978

E. Tahhan-Blttar?

CBU Lyon

1994

Figure 15. Some descendents of Denjoy

Arnaud Denjoy (see Figure 15) received a doctoral degree from Ecole NormaleSuperieure (ENS) in 1909. Denjoy was also on the committee of Kurepa, who appears in Figure 16.) R. de Possel also studied with Denjoy, and among his studentswas Roland Fraisse . See the obituary of Corominas by Pouzet [98] for details. Denjoy wrote about his student days in Mon oeuvre mathematique, sa genese et saphilosophe for a ceremony in March of 1971 in which he was awarded the Lomonssov medal in Moscow. The article is printed in a volume edited by Choquet [23] ,


which also includes a Notice necrologique by Henri Cartan. In particular, Denjoy wrote of mathematicians like Lesbesgue and Frechet who were coming on thescene ; and discussed a variety of subjects he was taught by Borel and Baire.

Ernest Corominas (see Figure 15) studied architecture and mathematics at theUniversity of Barcelona, and met J. Rey Pastorl". Corominas graduated in 1934.In 1939, because of the Spanish Civil War, Corominas went to Latin America wherehe was an architect in Chile, talked mathematics with Rey Pastor who broughthim to the University of Buenos Aires as an Assistant, and taught mathematicalfinance in Argentina. Then Denjoy offered him the position at C.N.R.S. in Francethat allowed him to pursue his doctoral degree there. (See [98] for details.) In histhesis [32], Corominas states that the beginning of his research was a problemposed to him by Rey Pastor (See [9] for more information on Pastor.) Corominasbecame interested in ordered sets during a year at the Institute for AdvancedStudies in Princeton; he lectured on them in Barcelona and worked on them inCaracas, before settling in Lyon where he attracted doctoral students to what hecalled "ordinal algebra."

Maurice Pouzet (see Figure 15) had Corominas as the head of the jury for histhesis, and Roland Frafsse as a member. Pouzet wrote of their influence:

I cannot separate their contribution in my work, I owe a lot to both. (If RolandFraisse and his work was the source of inspiration, Corominas taught me howto do mathematics) [99].

Elias Tahhan-Bittar (see Figure 15) did a master's degree with Di Prisco inVenezuela before continuing for the doctorate in France.

_ J. Hadamard

ENS Paris

1892

M. Frechet

ENS Paris

1906

D. Kurepa

U ParisSorbonne

1935

s. Todorcevicc

U Belgrade

1978

Figure 16. Some descendents of Hadamard

Jacques Hadamard (see Figure 16) received a doctorate in 1892 [50] from EcoleNormale Superiure (ENS) . He was one of the first two to prove the Prime NumberTheorem; he proved there are no (w,w*)-gaps in "t» ordered by eventual domination; and he [55] showed that geodesics on a surface of constant curvature havewhat is now called sensitive dependence on initial conditions, the key to chaos .J.P. Kahane has written about Hadamard for The Mathematical Intelligencer [67].Hadamard was not a prodigy (he ranked himself as a child in arithmetic as "ledernier, ou a bien peu pres" (see [22] and Kahane's article for further details),but he was what Paul Erdos called a dotegy, continuing to be active until over

19Pastor was a student of Caratheodory.

144 JEAN A. LARSON

ninety. Two contacts Hadamard made at Ecole Normale Superieure who continued to influencehim were his teacher Jules Tannery and physicist Pierre Duhem,a student a year above Hadamard (see pages 77-8 of [22]).

Maurice Frechet (see Figure 16) called himself the "first student and disciple"of Hadamard in a Notice necrologique [45] . He calls the Hadamard monograph [54]"prophetic" and quotes Hadamard on the first page of Les Espaces Abstraits, [46],the book for which Frechet is most well-known and which develops ideas from histhesis. Kendall [70] reminds us that Frechet introduced the notion of compactness.

The person for whom Kurepa trees are named is Buro Kurepa (see Figure 16).For a sense of the scope of his career, see the volume [65] of his selected papers,including his thesis, and the commentary on them.

Stevo Todorcevic (see Figure 16) said that Keith J. Devlin flew from Englandto Belgrade to be the external examiner of his dissertation . At Devlin's suggestion,Todorcevic, shortly after the defense, went to Jerusalem to listen to Shelah lectureon Proper Forcing. Todorcevic also said that he has profited from reading theErdos-Hajnallists [42], [43] of open problems. (See [120].)

_ A. Fraenkel

U Marberg 1914

A. Levy

HU 1958

M. Magidor?HU H. Judah

J B . BC1973 HU - . agarra-:

1987 UC Berkeley1991

Figure 17. Some descendents of A.A. Fraenkel

Abraham A. Fraenkel (see Figure 17) of Zermelo-Fraenkel fame spent timeas a student at universities in Munich, Marburg, Berlin and Breslau. His "Inauguraldissertation" at the University of Marburg on p-adic numbers is dated to1914 (see page ix of [13]), and was published in 1915 in Journal fUr die reine undangewandte Mathematik, which Fraenkel abbreviated on page ix of [13] as J. f.Math. His Habilitationsschrift came only two years after his Inauguraldissertation.Fraenkel worked with Kurt Hensel, who earned his doctorate in Berlin withLeopold Kronecker in 1884. Kronecker wrote his dissertation under LejeuneDirichlet in Berlin in 1845. Fraenkel also acknowledged the immediate influenceof Ernst Steinitz (see pages 96-97 of [48]).

Fraenkel taught at Marburg, spent one year in Kiel and taught most of hiscareer at Hebrew University (HU), where he took pleasure in his students:

Die vier hervorragendsten Schiller, die sich auf meinem Forschungsgebieteinen internationalen Namen erwarben (Y. Bar-Hillel, A. Levy, M. Rabin,2oA. Robinson), waren mir erst in Jerusalem vergonnt; auch das mir in Israel

20 See Figure 1.


beschiedene seltene Gliick, ein zw6lfjahriges "Wunderkind" zu entdecken, habeich in Deutchland nicht erlebt. (see p. 153 of [48] .)[The four brightest of my students (Y. Bar-Hillel, A. Levy, M. Rabin, A. Robinson), whom I first met in Jerusalem, later earned international reputations inmy research area; I had the rare luck to discover a twelve year old "wunderkind" , something I never experienced in Germany.]

Azriel Levy (see Figure 17) did his master's thesis in 1956with Fraenkel and wasawarded his Ph.D. in 1958 with Fraenkel and Abraham Robinson as his advisors .Starting at age seventeen [77] or eighteen [89], Robinson studied mathematics atHebrew University, where Fraenkel was his teacher. While he was a student there,he was an active member of the Haganah (an illegal Jewish defense organization)[89] . In 1939, he published his first mathematical article [106], On the independenceof the axiom of definiteness. After a few months of study in France on a scholarshipto the Sorbonne, Robinson went to England as the Germans invaded. In Englandhe worked during World War 2 on aerodynamics. In 1946 he was awarded a M.Sc.from Hebrew University and started teaching applied mathematics. He earnedhis Ph.D. at London University in 1949 with Paul Dienes with a dissertationon The metamathematics of algebraic systems. (See Macintyre [77] and Kochenin [133] for a discussion of Robinson 's contributions to mathematical logic; [133]includes discussions of his contributions to applied mathematics, arithmetic andphilosophy, and a brief vita.) Dienes studied in both Budapest and Paris beforeearning a doctoral degree at the University of Budapest (probably around 1905from the bibliography given in the memorial by R. Cooke [30]). Dienes continuedhis studies at the Sorbonne, earning a Dr. es Sc. degree. Both degrees were for workin complex anal ysis, but Dienes also worked on the algebra of infinite matrices andlogic.

When Menachem Magidor (see Figure 17) was about fourteen, he ran acrossin a used book store five thin volumes giving an introduction to mathematics inHebrew. They were written by Fraenkel in the late 1940's and were already out ofprint. The first volume was on number theory and algebra, the second on analysis,the third on set theory, the fourth on geometry and the fifth on topology. Theygave proofs that could be understood by a beginning reader and that conveyedideas of the different fields. Reading these booklets (he still has them), Magidorbecame fascinated with mathematics (see [80]) .

For a sample of expository writing about mathematics by Fraenkel [47], see theseries of articles he wrote for Scripta Mathematica, originally published by YeshivaUniversity and reprinted (in English) by the American Mathematical Society Press.Fraenkel was an editor for the journal, which was devoted to the philosophy, historyand expository treatment of mathematics.

Haim Judah (see Figure 17) earned a master's degree in 1983 with RolandoChuaqui (see Figure 5) at Pontificia Universidad Catolica de Chile under the nameof Jaime Ihoda before continuing graduate work with Magidor at Hebrew University.

Joan Bagaria (see Figure 17) started his dissertation with Haim Judah, andthen worked with Hugh Woodin (see Figure 10). Judah is listed as the advisor of

146 JEAN A. LARSON

record on the dissertation, and Hugh Woodin together with Robert Vaught andRalph McKenzie (both in Figure 6) complete the committee [11].

An article by Reuben Hirsch and Vera John-Steiner [63] in the Mathematical Intelligencer indicates the importance of contemporaries Lip6t Fejer (see thecommentary on Figure 20) and Frigyes Riesz (see Figure 18) in the flowering ofHungarian mathematics in the twentieth century. Tibor Rad6 (see Figure 18) andLaszlo Kalmar (see Figure 20) are also discussed in the article.

B. Szokefalvi-Nagy [115] reported that F. Riesz spent a few semesters at theEidgenosstsche Technische Hochschule in Zurich, a year in Gottingen where he wasinfluenced by Hilbert and Hermann Minkowski, and finished his university studiesin Budapest, where he was influenced by courses of Gyula Konig and J6zsef Kochk.In his doctoral dissertation on geometry, Riesz build on ideas of Frechet, (Thebrother Marcel Riesz of F. Riesz was also a well-known mathematician.) F. Rieszwas a founder of the field of functional analysis . See [107] for more informationabout Riesz.

F. Riesz

U Budapest

1902

T. Rad6U Szeged

1922

P. ReichelderferOhio State

1939

E. Fadell

Ohio State

1952

Figure 18. Some ancestors of Edward Fadell

Tibor Rad6 (see Figure 18) studied civil engineering at the Technical Universityin Budapest before enlisting in the army in 1915. He ended up spending four yearsin Russian prison camps, where the only books he could obtain happened to be onmathematics. After the war, he took up mathematics in Szeged. Later he movedto the United States and was chairman at Ohio State for a time. While Rad6 wasmainly an analyst, he also wrote on computable functions (see [100]) . For moredetails on his career, see [50] .

Paul Reichelderfer (see Figure 18) was an analyst like F. Riesz and T . Rad6.

_ Frank Wattenberg

1968, UW Madison Andreas Blass {1970, Harvard

Claude Laflamme?

1987, UM Ann Arbor

Carlos Montenegrof

1989, UM Ann Arbor

Figure 19. Some descendents of Edward Fadell


The trees of Figure 19 and Figure 18 connect, since Wattenberg was a studentof Edward Fadell. The primary field of both Edward Fadell (see Figure 18) andhis student Frank Wattenberg (see Figure 19) is topology, algebraic topology forFadell and fixed point theory for Wattenberg. Wattenberg also did nonstandardanalysis, hence knew a lot about ultraproducts and ultrafilters when he was theadvisor of Blass [20].

_ L. Kalmar

1926, U. BudapestA. Hajnal

1956, U. Szeged

Figure 20.

P. Komjath

1984, U. Budapest

At the time that Laszlo Kalmar (see Figure 20) was studying for his doctorate,there was no requirement of a formal thesis advisor. However, in a tribute to Lip6tFejer, Tandori stated the following'":

Fejer's pupils were, for example, Gyorgy Polya, Marcell Riesz, Otto Szasz,and then Jeno Egervary, Mihaly Fekete, Ferenc Lukacs, Gabor Szego, SimonSzidon, and from a later generation, Pal Csillag, Tibor Rado, and from aneven later generation, Pal Erdos , Laszlo Kalmar and Pal Turan. It comes toabout forty who obtained their doctoral degrees under him. He took pride inhis students [116] .

Fejer (doctorate in 1902), Kalmar and Komjath all got their degrees at theUniversity of Budapest, but the name changed over time, becoming PazmanyUniversity in 1921 and Eotvos Lorand University in 1950.

Hajnal [56] describes his first meeting (as a graduate student) with Paul Erdos,with whom he went on to write over 50 papers. See also [8] on their collaboration.Erdos and Hajnal [56] also worked with Milner and Rado (see Figure 22).

Next consider descendents and ancestors of Ronald Jensen.James Cummings (see Figure 21) told me that he spent time as a visiting

graduate student at Cal Tech working with Hugh Woodin (see Figure 10).Ronald Jensen (see Figure 21) was a student of Gisbert Hasenjager, In the

1950's, Kleene [71] and Hasenjager [57], building on Henkin's [58] proof of theCompleteness Theorem published in 1949, independently proved that if T is arecursive theory, then T has a natural number model in which the relations areAg. Kleene noted that Hilbert and Bernays [16] already gave this result in the casewhen T is a single sentence.

Hasenjager earned his doctorate at Miinster with Heinrich Scholz. Accordingto Jensen [66], Scholz was a professor of theology at Kiel, when around age forty,

21 In this quote all the names are listed in Hungarian, although some other referencesI have seen to M. Riesz are to Marcel; several individuals in the list have also publishedunder variations of their names in other languages . Matching each person with all theirnames can be quite interesting.

148

_ Ronald Jensen

1964, U. Bonn

JEAN A. LARSON

Adrian Mathiasf

1970, Cambridge

Charles Morgan"1989, U Oxford

Figure 21.

James Cummings"

1988, Cambridge

he became discouraged with philosophy and theology, since he felt that he had notseen real progress in either field. Then he ran across Russell's Principia Mathematica in the university library and became interested in logic. While at Kiel, he tooka degree in mathematics and was instrumental in bringing Fraenkel (see Figure 17)to Kiel. Later he held a chair in logic at Munster and established an Institute forLogic in the faculty of science there in 1944. Jensen said many German logiciansdescend from either Scholz or from K. Schutte, who was a student of Hilbert.

Jensen [66] said that Hasenjager scheduled his lectures for 4:00 p.m. partly inreaction to his experience listening to Scholz lecture at 7:00 a.m. in an unheatedroom after the war. Hasenjager's first contact with Scholz was as a schoolboy. In1937, Hasenjager volunteered to serve in the army with the idea of getting it overas soon as possible. Scholz got him assigned to the center for decoding-? whereScholz worked during the war, and after the war, employed him as his assistant inhis Institute for Logic.

Givant [51] connects Scholz and Tarski (see Figure 6) when he mentions thataround 1938, before Tarski went to Berkeley and got a position as a universityprofessor, that Scholz had told him a mathematician over forty without such aposition had bleak prospects.

Here is the final tree of the article.

Issai Schur

1901, U Berlin Richard Rado

1933, U Berlin

Figure 22.

Eric C. Milner?

1962, U London

22While Hasenjager and Scholz were working on decoding for Germany, a student ofChurch, Alan Turing, was decoding in England for the other side. (See Hilton[62] fordetails.)


Issai Schur23 (see Figure 22), in the curriculum vita accompanying his dissertation, lists twenty teachers and gives special thanks to Frobenius, Fuchs, Hensel(advisor of Fraenkel) and Schwartz.

After receiving a doctoral degree in Berlin, (Hitler came to power in 1933),Richard Rado (see Figure 22) went to Cambridge where he earned a second doctoral degree with G.H. Hardy. (See Rogers [104] for details) . Rado spoke of hisbeginning steps in mathematics in reply to a speech by L. Mirsky in which hepresented Rado with a volume dedicated to Rado for his 65th birthday

Having been registered as a research student at Berlin University for some time,I was in the habit of staring gloomily at a blank sheet of paper wondering howI could ever cover it with any worthwhile mathematics. One day I attendeda seminar conducted by a group of distinguished mathematicians. A fellowresearch student gave a lecture on van der Waerden's theorem on arithmeticalprogressions. To me the theorem sounded quite incredible and the proof a stringof fallacies. I went away determined to shatter whatever belief there could existin the truth of such a theorem . But on studying the matter more closely I hadto admit that the theorem was true and the proof sound. This gave me mystart in mathematics and I have never looked back [104].

6. Final Remarks

I was able to find out the thesis advisor of the twenty-five individuals with doctorates who attended the conference in Curacao and the fifteen individuals withdoctorates attending the conference in Barcelona (Bagaria and Di Prisco attendedboth) . Thus I have traced back forty-one individuals, including Jorge Martinez,director of the Caribbean Mathematics Foundations, Peter Komjath and two coauthors of papers in the volume who did not attend either meeting: Carl Darbyand Saharon Shelah . There were a total of fourteen connected components in theresulting trees. Of the fourteen individuals working in set theory, logic and foundations in the United States over the period 1995-7 (Baumgartner, Cummings,Darby, Henle, Kechris, Kojman, Koszmider, Larson, Laver, Miller, Mitchell, Roitman, Schimmerling, Woodin), a total of four were descendents of Church, an additional one was a collateral relative of Church, five were descendents of Tarski andthe remaining three (Cummings , Henle, Roitman) were related to neither Churchnor Tarski. This computation lends credence to the assertion that was the original impetus for developing this paper. Of the forty-one individuals in the sample,seven are descendents of Church, three are collateral relatives of Church, nine aredescendents of Tarski , four are descendents of Hilbert, twenty-three are unrelatedto Church , Tarski or Hilbert, as far as I have been able to discern.

In the process of building this genealogy, I discovered a chain of five currentlyactive individuals linked by the advisor/student relationship: Mac Lane, Solovay,McAloon, Dehornoy, Sureson.

23See Ledermann [76], where there is also a list of doctoral students of Schur.

150 JEAN A. LARSON

In talking to people about the advisor-student relationship, I have discoveredthat it can take many forms. Sometimes it is a formal role in which the advisorvouches to the university for work the student has done with very little input fromthe advisor . Other times, the thesis advisor guides the student through suggestionson papers to read and problems to work on, by teaching both techniques andbackground material of the field, and through direct comments and suggestions onthe thesis work as it develops.

Some students, like Cummings and Roitman, visit universities other than theone in which they are enrolled; with email there are increasing collaborationswith distant people. Some students have multiple official advisors, like Elgueta,Koszmider and Levy.

Thesis topics and knowledgeof the field may come from interaction with fellowstudents or visiting researchers, as in the case of R.L. Moore and Oswald Veblen[121]. Baumgartner [20] writes of a visit to Berkeley in 1964by A. Hajnal where helectured on Erdos-Rado set theory (see [56]). In the audience was Silver who thenshowed in his thesis that the existence of 0# follows from that of Erdos cardinals .

For those who would like to look further, A. Kanamori [68] has brief historyof set theory. J . Dawson [35], in his book on Godel, has included biographicalsketches of a number of individuals mentioned in the paper. Dissertation AbstractsInternational [29] proved to be an excellent source of information on those whosedegrees are earned in the United States; other sources for dates of degrees include[1], [84], [50], [61], and [21].

In addition to the references indicate so far, I would like to recommend twoweb sites:

The Theoretical Computer Science Genealogyhttp://hercule.csci.unt.edu:80/ genealogy/

The MacTutor History of Mathematics archivehttp://www-groups.dcs.st-and .ac.uk:80/ history/index.html

ACKNOWLEDGMENTS: I am grateful for the hospitality of Instituto Venezolanode Investigaciones Cientificas in Caracas, Leeds University, Mathematisches Forschungsinstitut Oberwolfach, and the University of California at Berkeley, wherepart of this research was completed. I would like to thank the many individuals helped with this project, including librarians, especially at the Universityof Florida; those who shared stories of their mathematical beginnings or information their mathematical ancestors and/or colleagues (Addison, Bagaria, Dehornoy, Goldstern, Hajnal, Jensen, Just, Krawczyk,Kojman , Komjath, Koszmider,Magidor, Marek, Pouzet , Roitman, Scott, Steel, Todorcevic, Vaught, Woodin, Zakrzewski). Donald Burkholder and Murali Rao determined that Walsh was the advisor of Doob and confirmed it by calling Doob. Patrick Bonace looked in recordsat Ohio State to find the advisors of Fadell and Reichelderfer there . Hans-DieterDonder, Peter Koepke, Irene Hueter and Gerard Emch helped me with resources inGerman. Wilfrid Hodges shared with me a mathematical context for Hasenjager.Stevo Todocevic encouraged me to include stories of mathematical beginnings andmade suggestions about early versions of the paper .

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RECURRENT POINTS AND HYPERARITHMETIC SETS

A.R.D.MATHIAS!

Centre de Recerca MatemdticaInstitut d'Estudis CatalansApartat 50E-08193 BellaterraEspana

Abstract.We give an example of an iteration with recursive data which stabilisesexactly at the first non-recursive ordinal. We characterise the points in the finalset as those attacked by recurrent points, and use that characterisation to showthat recurrent points must exist for any iteration with recursive data which doesnot stabilise at a recursive ordinal.

O. Introduction.

This paper is the third of a series that studies the closure ordinal O(a, f) of aniteration defined using a continuous function f from some Polish space X to itself,starting from a point a EX. We summarise the definitions:

Definition 0.0. wf(x) is the set of those points y E X such that each neighbourhood of y contains for each n a point fm(x) for some m > n . It is not excluded thatfm (x) = fP (x) for some p ~ n; thus periodic points of the form fm (x) are countedas belonging to wf(x). We write x ("\.f y if y E wf(x), and omit the subscript f indiscussions for which f has been fixed. We read x rv y as "x attacks y".

Proposition 0.1. (i). wf(x) is a closed subset of X .[ii). If x rv y and y r» z then x rv z.

lThe address given here is the one where I wrote the paper. A current email addressis [email protected].

157


158 A.R.D.MATHIAS

Definition 0.2. For A ~ X, set r(A) =df U{wf(x) I x E A}. Then using thisoperator and given a point a E X, define recursively sets AV(a) = AV(a,l):

AO(a) =AI3+1(a) =

AA(a) =

wf(a)

r(A.B(a))

nAV(a)V<A

for A a limit ordinal

Proposition 0.3. (i). For all ordinals a, {3, a < {3 =::} k"(a):> A.B(a).(ii). x E AV(a) =::} f(x) E AV(a).

(iii). For each ordsnal u, A/L(a) = wf(a) n nV</L AV+l(a).

Definition 0.4. Let fJ(a,1) =df the least ordinal fJ with AIJ(a) = AIJ+l(a);AOO(a, I) =df AIJ(a,f); E(a, I) =df wf(a) <, AOO(a, I). Points in AOO(a, I) aresaid to abide, and points in E(a, I) are said to escape.

In the first paper [7], we linked escape to well-foundedness by establishing anequivalence of a kind familiar to students of monotone inductive definitions:

Proposition 0.5. A point Z E wf(a) abides if and only if there is a sequence Zi(i < w) such that Zo = Z and for each i a 0. Zi+l 0. Zi.

We used that link to show that for any X, f and a, fJ(a, I) was at most the firstuncountable ordinal. In the second paper [8], we gave a method of placing points atthe nodes of a well-founded tree, where the trees in question were all subtrees of 5,the set of finite strictly increasing sequences of odd prime numbers (excluding 1),including 0, the empty sequence; and we showed, using the well-foundedness of thetree, that for a particular f the method would permit the construction of a pointa with fJ(a, I) equalling the rank of that tree, which might be any pre-assignedcountable ordinal .

Two spaces were studied: the Baire space N = W W of all infinite sequences ofnatural numbers, and the Cantor space W2: in the first the function f consideredwas the (backward) shift function s defined by s(b)(n) =b(n + 1) for b : W ~ w,and in the second it was S3 defined by s3(b)(n) = b(n+ 3) for b : W ~ {O, I}.

But the method of assignment of points was independent of the tree underconsideration, and in particular did not rely on the trees being well-founded; andin the present paper we apply it to ill-founded trees and obtain this result :

Theorem 0.6. There is a recursive point a in Baire space N such that fJ(a,s) =wfK, the first non-recursive ordinal.

We now review that method: for each s E 5 we shall define a point X s in Bairespace, by induction on the length of s, so that

We start by settingx0 = 0,4,8, .. .

so that X0 attacks nothing.

Recurrent points and hyperarithmetic sets 159

Suppose now that for some s E S, we have defined xs , and that t is an immediate extension of s, so that t = s~(k}, where k is an odd prime exceeding allthose occurring in s. Write 1rt for the product of all the primes occurring in t, sothat 1rt is a square-free number of which k is a factor. For each natural number nwe write 1rt,n for the number (1rt)n+l. Note that for non-empty rl and r2 in T, 1rr 1

divides 1rr 2 iff r2 ~ rl. We then define

where the positive integers nt,i are chosen strictly increasing and such that thepredecessors of each occurrence of a power of 1rt in Xt (after the first) form astrictly increasing sequence of even numbers: to enable that choice to be made wemaintain inductively the property that the X s have infinitely many even numbersin their range .

There is considerable freedom in the choice of the integers nt,i, and a specificrecursive choice is given in Long Delays. To prove our main theorem any recursivechoice that guarantees the truth of the various lemmata in §1 of Long Delays willdo: as before, when we are discussing the shift function s acting on Baire space.N we write b t> C to mean that for some non-negative integer n, b = sn(c). Weshall, though, see in Proposition 4.4 that a more careful choice of nt,i will give anexample with rather sharper details.

Now let T ~ S be a tree which contains the empty sequence 0 and which isclosed under shortening. By a process recursive in T we define a point XT of Bairespace with the property that

sET ==> XT rv X s '

The definition differs slightly from that in Long Delays: there we listed allbottom points of a well-founded tree ; here, where we consider trees that mighthave no bottom points, we must list all points of the tree . In fact, in the interest ofuniformity, we list all members of S recursively as (s, liE w) so that each occursinfinitely often, and then proceed as in Long Delays to define

where the integers nT,i are chosen strictly increasing, and such that the immediatepredecessors of the occurrences of numbers n == 2 (mod 4) are distinct positivemultiples of 4, and t i is the first Sj in sequence after previous tk'S to be a memberof T: so, in effect, we always check to see whether Sj E T, and if it is not , we donothing at that stage but proceed to the next.

With the above definitions we have the following

Proposition 0.7. If T is recursive so are the point XT and the sequence

(Xs Is E T) .

160 A.R.D.MATHIAS

We consider what happens when there is an infinite path through T , and findthat our principal technical lemma, Proposition 3.0, which we prove in full generality for arbitrary Polish spaces and continuous functions, implies that, in the caseof our systems of points on trees there is at the end of each such path a recurrentpoint which attacks every point along it.

In §1 we review some familiar material on recurrent and minimal points. In§2 we use Proposition 0.5 and arguments from analysis to characterize abidingpoints as those attacked by recurrent points. In §3weformulate as Proposition 3.0the construction underlying our characterization, derive some corollaries and withthe help of a set-theoretic argument establish the existence of maximal recurrentpoints. In §4 we apply the ideas of §3 to the context of trees T ~ 5, pointsXT , and the shift function 5. In §5, we review some material from the theory ofhyperarithmetic sets, in §6 we complete the proof of Theorem 0.6, and in §7 weprobe further links between hyperarithmeticity and our dynamical context , anddiscuss some open problems.

1. Minimal points.

Definition 1.0. A recurrent point is a b such that b rv b.

It has long been known that the existence of recurrent points is neither certainnor impossible:

Example 1.1. Let X =R, and f(x) == x + 1. Then f has no recurrent points.

Theorem 1.2 (AG). Let X be a compact Polish space and f : X ~ X continuous. Then recurrent points exist.

Remark 1.3. The above use of AG could be reduced to an application of DG byworking in L[a, f) and appealing to Shoenfield's absoluteness theorem.

We may use the following lemma since in a metric space first countability andseparability are equivalent conditions.

Lemma 1.4 (AG) . In a first countable space X there can exist neither a strictlydescending sequence (Gv I u < WI) nor a strictly ascending sequence (D v I v < WI)of non-empty closed subsets of x .Proof. Given a descending counter-example in a space with countable basis {Ns ISEW}, pick Pv E O; \ Gv+! , and Sv E W with Pv ENs" and Ns" n Gv+! empty.There will be v < 8 < WI with Sv = so. But then Po E GOnNS6 ~ Gv+! nNs" =0,a contradiction.

In the ascending case, pick Pv E Dv+! \ Dv, and Sv E W with Pv ENs"and N s" n D II empty. Again there will be v < 8 < WI with SII = So . But thenPII E D II+I n N s" ~ Do n N S6 =0, another contradiction. -j (1.4)

Remark 1.5. Hausdorff in §27 of his book Mengenlehre proves with a beautifulargument that, more generally, there cannot be an uncountable sequence, whetherstrictly increasing or strictly decreasing, of sets that are simultaneously F" andGo: in more recent notation, such sets are called ~g, while an F" set is called Eg,and a Go set ng.

We shall use Hausdorff 's characterisation of ~g sets in §7 to see that while theset ofrecurrent points is always rrg it need not be Eg.Proof of Theorem 1.2. We know that each w/(x) is a closed set, which, bysequential compactness is non-empty, and that if y E w/(x) then w/(y) ~ w/(x) .Start from x, and set Co = w/(x). We shall define a shrinking sequence of closedsets all of the form w/(z) .

If C~ = w/(x~) ask if there is ayE C~ such that w/(y) is a proper subset ofC~ : if not, then x~ is recurrent (in a strong sense, indeed). If there is, pick somesuch and call it XHI , and take C~+I =w/(xHt} .

At limit stages, take the intersection, call it C~: by compactness it will benon-empty. Pick x>. in it. Then for each v < A XII r'\. X>.j so w/(x>.) ~ C~. SetC>. = w/(x>.) and continue .

By the Lemma this process breaks down before stage WI: when it does, we havereached a z such that VwEw/(z) w r'\. z: since w/(z) is non-empty, such a z isevidently recurrent. --j (1.2)

Remark 1.6. In such a case the set w/(z) , or the point z, is called minimal.

The minimal sets are pairwise disjoint closed sets, which might, but need not,partition the set of recurrent points. Here are some examples.

Example 1.7. Let X = [0,1] and f(x) == x2: then Aoo = {O, I}, although f is

onto and 1-1 and X is compact. The minimal sets in this case are {O} and {I} .

Example 1.8. Let X = {x E R 21 d(x, 0) ~ I} . Let f be rotation by an irrational

multiple of 211". The minimal sets are the concentric circles with centre the origin,so they partition the space, in which every point is recurrent.

Example 1.9. A case where a non-minimal but recurrent y attacks two inequivalent minimal points: let y E 4w attack everything in that space, let a have only O'sand 1's, and let b have only 2's and 3's. By compactness, a and b, if not minimalthemselves , will attack minimal points. Neither a nor b can attack y.

Remark 1.10. In a compact space, the set of all points in some minimal set is{x I Vy X rv Y => Y rv x}, so it is ITt. Hence by a theorem of Burgess eitherthere are at most NI minimal sets or there is a perfect set of inequivalent minimalpoints, where points x and y are considered equivalent if x rv y rv x.Proposition 1.11. Suppose no image of x is recurrent. Then w/(x) is nowheredense.

Proof. Each point of w/(x) is a limit of points (namely the Jk(x), for k E w) notin that closed set, hence its interior is empty. --j (1.11)

Corollary 1.12. Let r be Cohen generic over L[f, x] and suppose that no imageof x is recurrent. Then r is not attacked by x.

Proposition 1.13. Suppose f is 1-1 and C is a minimal class which is not meagre. Then C =X.

Deny, let t be on the boundary of C and v in the interior of C. t rv V byminimality, and so f(u) is near v (and therefore in C) for every u ¢ C sufficientlynear t. But each point in C is f(w) for some win C, as everything is zapped. Thiscontradicts f being 1-1. --j (1.13)

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2. Recurrent points

A.R.D.MATHIAS

The following result shows for every Polish X, continuous f and point a that,provided some point in wf(a) abides, recurrent points exist. We emphasize thatthe space is not assumed to be compact. The apparent use of the Axiom of Choiceis avoidable.

Theorem 2.0. Let X be a complete separable metric space, f : X --t X a continuous map, and a, x arbitrary points in X . Then

x E Aoo(a, J) ¢=:::} 3b a r+ b rv b r+ x .

Proof. We know from Proposition 0.5 that for any x, x E Aoo(a, J) if and onlyif there is an infinite sequence Xi with a rv Xi+! rv Xi r+ Xo =X for each i, Henceif a rv b rv b rv x, the point X is in Aoo, as we could take Xi = b for i > O. Inparticular every recurrent point is in A00 •

For the other direction we use 0.5 again to suppose that for each i < w, a r+Xi+l r+ Xi. Our task is to build a recurrent b with a rv b rv Xo.

We shall define a sequence of points Yi starting with Yo = Xo and convergingto a point b, such that for each i, a r+ Yi and b r+ Yi. Since wf(a) and wf(b) areclosed, that will give a rv b and b rv b, so b is a recurrent point with a rv b rv XO .

To define the sequence Yi we shall define various sequences of positive reals tending monotonically to 0, and we shall define various strictly increasing sequences ofpositive integers.

More specifically, for each i < W we shall define a sequence (c~h<w of positivereals tending monotonically to 0, and for 0 < i < W a strictly increasing sequence(t'Ok<w of natural numbers. Further we shall define a decreasing sequence (T/i)i<wof positive reals tending to 0, and we shall define a strictly increasing sequence ofpositive integers (mih~i<w.

Our definition takes place in infinitely many rounds. In Round 0, we shall definethe point Yo, the sequence (c~) and the positive realT/o. In Round 1, we shall defineml , Yl, the sequences (11) and (cD and the positive real m . For n > 1, we shallby the end of Round n-l have defined mn-l, Yn-l, l~-l and c~-l for each k, andT/n-l, and in Round n we shall define m n, Yn, lk" ck' and T/n'

Let 1IJ(i, n, 7, k) be the statement that

17 - Ynl < ck' ::::} Ifl;:+l:-l+...+l~ (7) - Yi-ll < c~-l

We shall verify in Round n, for n ~ 1, that

VkV7Vi((k E W &7 EX & 1 ~ i ~ n) ::::} 1IJ(i, n,7, k)).

In fact , for each 7 and k, 1IJ(n, n,7, k) will follow from our choice of c~ and t'~ ; andthen the other cases will be covered by the following

Lemma 2.1. If ck' and lk' have been defined, then for i < n,

(1IJ(n, n, 7, k) & 1IJ(i, n - 1, fl;: (-y), k)) ::::} 1IJ(i, n, 7, k).

Proof. Let I'Y - Ynl < e~ . By \l1(n,n,'Y,k), Ifli:(-y) - Yn-ll < e~-l: so we mayapply \l1(i, n - 1, fli: (-y), k) and use the fact that

fl~-I+...+ti (li: (-y)) =li:+l~-I+ ...+l~ (-y)

-l (2.1)We are now ready to begin our construction. In it, we shall use without further

comment the general lemma that if c ("\, d then f (c) rv d and c rv f (d).

Round O. Put Yo = Xo, choose an arbitrary sequence e~ of positive reals tendingmonotonically to 0 as k --t 00, and set 1'/0 = teg.

Round 1. Pick ml such that If m 1 (xd - Yol < 1'/0 : that is possible as Xl rvYo = Xo· Put Yl = ["" (Xl)' Choose a sequence eA < et < e~ < .. . such that

Vk If l l (YI) - Yol < ~e~: that can be done as Yl rv Yo .Choose a sequence el tending to 0 monotonically from above such that

that can be done as fll is continuous at Yl.That implies that for all k E w and all 'Y EX,

that is, that \l1(I, 1, 'Y, k) . Set 1'/1 =min(~eg, teD =min(~T/O, teD·

Round 2. Pick m2 > ml such that If m 2 (x2) - Yll < 1'/1 -possible as X2 rvYl- and put Y2 = fm 2 (X2)' Choose a sequence £~ < e~ < e~ < .. . such that

Vk Ifl~ (Y2) - Yll < ~el : that can be done as Y2 rv Yl·Choose a sequence e% tending to 0 monotonically from above such that

VkV'Y EX (I'Y - Y2! < e% ~ Ifl~ ('Y) - fl~ (Y2)1 < ~e~) :

that can be done as fl~ is continuous at Y2 .That implies that for all k E w and for all 'Y EX,

and therefore2 12+11 0

I'Y - Y21 < ek ==> If k k (-y) - Yol < ek

which are the statements \l1(2, 2, 'Y, k) and \l1(I , 2, 'Y, k) respectively. Set 1'/2 te~) =min( ~1'/1' te~) and continue to the next round.

Round n, for n > 2. Pick mn > mn-l such that Ifmn (xn ) -Yn-ll < 1'/n-l, and putYn = l"" (xn). Choose a sequence er < e~ < .. . such that Vk Ifli: (Yn) - Yn-ll <~e~-l: that can be done as Yn rv Yn-l .

Choose a sequence e~ tending to 0 monotonically from above such that

164 A.R.D.MATHIAS

that can be done as flk is continuous at Yn'That implies that for all k E W and for all 'Y E X

h - Ynl < e~ =? If lk ('Y) - Yn-ll < e~-l

which is the statement '11(n, n, 'Y, k); we have seen that it follows from statementsestablished in previous rounds that for n > i ~ 1,

I'Y - Ynl < e~ =? Iflk+l~-1+...+l1 ('Y) - Yi-ll < e~-l

which is '11(i,n,'Y,k). Set TJn = min(~TJn-l' te~).

Once all the rounds have been completed, we shall have defined a sequence Yisuch that for each i, IYi+l - Yil < TJi· By definition TJi+l = min(~TJi, te~t~), soin particular TJi+l ~ ~TJi' and so Ei<w TJi is convergent. Hence (Yi) is a Cauchysequence, and hence by the completeness of the space X is convergent. Let b beits limit.

Lemma 2.2. For each k, Ib - Ykl < eZ.Proof. For each k, TJk ~ teZ, TJk+l ~ ~TJk ~ kez, and so for each j ~ k,TJi ~ 2k- 2

- ieZi we know that for each i, IYi+l - Yil < TJi, and hence for k < i,IYi - Ykl < TJk + .. . + TJj -l; thus

Ib-Ykl ~ LTJi ~ (~+ ~ + . . .)eZ = ~eZ ·i~k

-j (2.2)Fix i. We assert that b rv Yi. Thus, we must show that

"Ie> 03n Ir(b) - Yi! < e;

moreover that n may be chosen arbitrarily large.Fix e> O. Pick k > i such that 4 < e. By the lemma, Ib - Ykl < eZ, and so

applying'11(i+l,k,b,k),

Ifl~+l:-1+...+l1+1 (b) - Yil < 4 < e,

as required.Note finally that as k can be chosen arbitrarily large, the power of f applied

to 6, which is at least l~+l, can also be made arbitarily large.

Our theorem is proved. -j (2.1)

3. Maximal recurrent points.

In fact our proof of Theorem 2.0 has established the following statement:Proposition 3.0. Given X, f, and a, suppose that for all i a rv Zi+l rv Zi rv. .. rv zo. Then there are natuml numbers rno < rnl < ... such that settingYi = fm; (Zi), the sequence (Yi) is convergent with limit b, say, and b rv uc for eachi , It follows that b is recurrent, and that for all i a rv b rv Zi and Wj(Zi) = Wj(Yi).

Remark 3.1. Note that if the points z~ form a secondset satisfying the hypothesisof the Proposition, with Vi Zi rv z~ rv Zi, and Y~ , b' are the outcome of repeatingthe argument, then

and so b rv b' rv b.

Proposition 3.2. In that context, wf(b) is the closure of UiWf(Zi).

Proof. Write Gb for wf(b), C, for Wf(Zi), and G for the closure of Ui Gi. Each C,is closed topologically and also under the action of l, hence so is G. Gb is a closedset containing Ui Gi, and therefore Gb ;2 G. But bEG, being the limit of thesequence Yi, so each fk(b) is in G, and therefore each point of Gb is in G. -l (3.2)

Definition 3.3. Call a point b maximal recurrent in wf(a) if a n. b rv b andwhenever a n. c rv c rv b, then b rv c.

With the help of the axiom of choice the above proposition yields the following

Corollary 3.4 (AG). If d is a recurrent point in wf(a) , then there is a point bwhich is maximal recurrent in wf(a) with an. b rv d.

Proof. Set do = d. If do is not maximal in wf(a), pick dl with a rv dl rvdl rv do tj\. dl ; if dl is not maximal, continue. The proposition tells us that ourconstruction can be continued at countable limit ordinals. If we never encountera maximal recurrent point, then our construction will yield for every countableordinal 11 a recurrent point dll with a rv d( rv dll r/+ d( for 11 < ( < WI' But thenthe sequence (wf (dll ) I 11 < WI} will form a strictly increasing sequence of closedsets of order type WI, contradicting Lemma 1.4. -l (3.4)

Remark 3.5. Again, that use of AG could be reduced to an application of DGby working in L[a,fJ and appealing to Shoenfield's absoluteness theorem.

Remark 3.6. We could also formulate the notion of a maximal recurrent pointin the space X as a whole, without reference to a particular point a; the sameargument will prove that if recurrent points exist, so do maximal ones. In a casesuch as the shift function acting on Baire space, the maximal recurrent points willbe simply be those whose orbit is dense in the whole space.

4. Points at the end of paths

We apply the results of §3 in the context of §O. We are in Baire space and considerthe shift function s, We have a tree T ~ S which is closed under shortening andwe have defined points X s for sET, and a point XT.

4.2. First, suppose that we have an infinite path p through T.Set ai = Xpti. Then by construction aiH n. ai; as in §3 we may find integers

mi such that if we set Yi = fm; (ai), the sequence (Yi) will be convergent to arecurrent point we shall call xp ' There is likely to be freedom in our choice ofintegers mi, so that we do not know that xp is uniquely determined by the pathp. However, by 3.2, wf(xp ) is uniquely determined, and we know that for each i,

166

and that for any point z,

A.R.D.MATHIAS

if Vi XT rv z rv Xpti then Vi z n.. Yi and hence z rv xp '

Proposition 4.1. We may adjust our choices of the various integers employed sothat xp is always recursive in p.

Proof. By inspection of the argument of §2. -I (4.1)

Henceforth xp may be taken to be {e}P for the least index e such that the eth

function recursive in p is a possible value for xp '

Now we may use lemmata from Long Delays to prove:

Proposition 4.2. If XT rv (3 then either there is a uniquely determined infinitepath p throughT such that XT rv xp rv xp n.. (3, or there is an sET with (3 t> X S '

Proof. Suppose that XT rv (3. Any odd number that occurs in (3 is of the form1I"t,n' Suppose that (3(i) = 1I"t,n and (3(j) = 1I"s,m are odd numbers occurring in (3,then, taking k > max{i, j}, and applying LD 1.10 to a vET with (3 r k C XII'

we see that both s and t must be initial segments of v. Hence the t's such that apower of 1I"t occurs in (3 define a path through T which may be empty, or finite, orinfinite.

If that path is empty, or, in other words, if no odd number occurs in (3, then(3 t> X0, by LD 1.12 and the fact that no number occurs twice in X0. If that path isnon-empty but finite, then there is a longest s such that some power of 11"s occurs,and then by LD 1.13 (3 t> X S ' If the path is infinite, let us call it P/3. Then xPf3 rv (3,since given k there is an l such that (3 r k C Xpti, and x p n.. Xpti ' -I (4.2)

Remark 4.3. Thus, in this context, if (3 defines P/3 ,

X Pf3 ~Thrjng P/3 ~Thrjng (3.

The above proposition, coupled with some facts about hyperarithmetic sets,is enough for the proof of the main theorem of the paper. We pause to prove arefinement.

Proposition 4.4. Given any subtree T of S closed under shortening, the numbersnt,k (for t ~k E T) may be chosen so that whenever XT rv (3 and (3 defines aninfinite path p through T, (3 rv X s for each s E p, and therefore (3 rv xp , and (3 isrecurrent.

We ensure that whenever t = s~(k), the integers nt,iare chosen so that foreach s' ~ s, X s' r lh(t) C Xs rnt,i, in addition to our earlier requirement, set outin Long Delays, that the numbers xs(nt,i - 1) immediately preceding each powerof 1I"t in Xt should form a strictly increasing sequence of multiples of 4, and the(new) requirement that xs(nt,i) will always be a power of 1I"s.

If we have done that, then we may show that for sET, (3 rv X!. For let N begiven, and pick t of length at least max(N,lh(s) + 1) for which some power of 1I"t

equals (3(a), where a > N. Let c be the least integer exceeding a such that (3(c)

is a power of 7r'U for some u with u -< t. Let b be the largest number less than cfor which (3(b) is a power of 7rt , so that a ~ b < c. Let s = t t (lh(t) - 1). Then{3 t c + 1 C X w say, and so the segment (3 t [b + 1, c) equals X s t nt,m for some m,and hence X§ t N is a segment of {3 t [(3 + 1, c) and thus is a segment of (3 startingafter stage N . -l (4.4)

Remark 4.5. Generally, by §2,

In our tree context, for given {3 not near or attacked by any x s , the recurrent'Y = xPf3 that attacks {3 is recursive in (3. This gives us an easy way of showing that9(XT'S) is countable.

Let 21. be the set of nodes s such that the tree below s is ill-founded. 21. is ofcourse a countable set of finite sequences , and therefore codable by a single real,a.. Thus we have

which is Eg in a and XT . Aoo is thus Borel and the ordinal9(xT's) countable.If in addition we have chosen the integers nt,k as in 4.4, then Aoo will have this

yet simpler characterisation:

AOO ={{313sE 21. {3 e- xs } U {{31 XT rv {3 rv {3},

making Aoo the union of an E; and a G6 set .

5. Hyperarithmetic points and closed sets

5.2. There is a countable family of functions from w to w called the hyperarithmetic functions (or HYP for short) which are the trouble-makers when it comesto inductive definitions.

The paper The next admissible set by Barwise, Gandy and Moschovakis [2]lists nine equivalent definitions of this family, of which we state four. The readerwill find much more information than we can give here in the treatises of Barwise[1], of Mansfield and Weitkamp [6], of Moschovakis [9], of Hartley Rogers, Jr. [12],and of Shoenfield [13].

(5.2.0) A function 0: w -t w is HYP iff its graph {{m, n} Ioem) =n} is ~i;

Since for a total function 0 : w -t w, oem) = n ¢::=} Vk(k t= n => oem) t= k),it is sufficient to require that the graph be Et.

(5.2.1) A function 0 : w -t w is HYP if and only if it is a member of everyw-model of analysis;

From that definition it is plain that [o lois HYP} is ITt.(5.2.2) A function 0 : w -t w is HYP if and only if it is recursive in some He,

for e a recursive well-ordering;Here He is the hierarchy defined by Kleene proceeding by Turing jump and

effective union.

168 A.R.D.MATHIAS

(5.2.3) A function a : w 4 w is HYP if and only if it is a member of thesmallest admissible set containing w + l.

Some comments on that last definition. Let us write V'R for the smallest transitive model of Kripke-Platek set theory including the axiom of infinity. V'R = LwCK,

1the initial segment of the Godel constructible hierarchy up to the Church-Kleeneordinal, the first non-recursive ordinal, wfK.

V'R is the collection of all sets coded by a HYP well-foundedextensional relationon w. It is well-known that V'R is not a ,a-model; that is, that there are linearorderings in V'R which are not well-orderings but which V'R believes to be wellorderings. Such linear orderings we shall call, following Harrison [3], pseudo-wellorderings .

5.3. In particular it follows from the Kleene Boundedness theorem that there existrecursive pseudo-well-orderings: for by that theorem the set of indices e of recursivewell-orderings is a ITt set, but not Et; the set of those indices e of recursive linearorderings with no HYP descending chains may be seen to be Et by using definition(5·0·1), and includes all the indices of recursive well-orderings; and so there mustbe an e in the second set but not in the first.

5.4. The proof of the celebrated Cantor-Bendixson theorem, that every closed setis the union of a perfect set and a countable set, starts from a closed set C andproceeds by iterating the operation of taking the derived set; the sequences of setsCV is a descending sequence of closed sets and so by Lemma 1.4 stops at somecountable ordinal, called the closure ordinal of the construction. Call Coo the finalset: it equals its derived set (otherwise the sequence would continue to shrink)and so Coo is a perfect set, that is, is closed and has no isolated points . If C iscountable, Coo will be empty, otherwise Coo will be of cardinality the continuum.Each CV <, cv+l is countable (or finite). So C <, Coo is countable, and we haveshown that every closed set is the union of a perfect set and a countable set .

That proof was analysed by Lorenzen and by Kreisel [5] in the context of V'R.For a precise statement of their results the reader should consult the review [11]by Moschovakis. For our purposes we note the following:

Theorem 5.3. Let C be a recursive tree defining a closed set. Let O" denotethe Cantor-Bendixson sequence, and Coo the perfect kernel. Let 8 denote the firstnon-recursive ordinal.

(5.4.0) If art HYP, a E [COO] <=> a E [C].(5.4.4) If a E HYP, a E [COO] <=> a E [OS].(5.4.4) Coo =Co.

Related to that is the following curious compactness phenomenon at 8, whichholds even if the space we are thinking about is not compact:

Proposition 5.4. If C is recursive and for each recursive ordinal CV is nonempty, then Coo is non-empty.

Proof in brief. The hypothesis implies that for each e E W 0 there is an x andan e-CB-frame for x , namely an array of points that witnesses the survival ofthe Cantor-Bendixson process for lei steps . But that is a Et statement, which by


Kleene must be true of some pseudo-well-ordering: so there is an x and a pseudoframe for x . But that gives x E Goo . -l (5.4)

Corollary 5.5. Let G be a countable recursive closed set. Then there is a recursiveordinal v such that [G"] = 0 .

All its members are HYP, so Goo is empty so some G" is empty for a recursivev. . -l (5.5)

6. Proof of our main result.

Consider a non-empty recursive linear ordering which is ill-founded but which hasno HYP descending paths: such exists by 5.1. Form the tree of all finite sequencesof strictly decreasing sequences in that ordering, ordered under end-extension. Asdescribed in Long Delays, that may be copied to a tree T ~ S. We work with thatlatter tree. It is recursive and ill-founded but has no HYP descending paths.

We build points x s , for sET, and XT .

Proposition 6.0. O(XT'S) =wfK .

Proof. As proved in Harrison's thesis, the said linear ordering will have a wellordered initial segment of length exactly wfK. Hence the tree T will have pointss below which it is well-founded, and the ranks of such s will be exactly the set ofrecursive ordinals. So by the arguments of Long Delays, O(XT,s) ~ wfK •

Now suppose XT r.. {3. If {31> X s where SET, {3 will be HYP, indeed recursive,as each X s is recursive. Given our choice of X0, there will be no points attacked bysome X s that are not near some Xt with shorter t. If the tree below s is ill-founded,{3 will abide, otherwise, if the tree below s is well-founded, then {3 will escape, andthe ordinal at which it escapes will be recursive.

So the only case remaining to be discussed is when {3 defines an infinite pathPf3 through the tree: in this case, since Pf3 is recursive in {3, and, by choice of T,Pf3 cannot be HYP, {3 cannot be HYP. By §4, xPf3 rv {3: since xPf3 is recurrent,{3 E Aoo.

So every point that escapes does so at a recursive ordinal, yieldingO(XT'S) ~ wfK. Hence O(XT'S) = wfK. -l (6.0)

Theorem 0.6 follows immediately from 6.0. -l (0.6)

Remark 6.1. If we make the more refined choiceof nt,i's sketched in 4.4, we shallhave the following exact picture of Ws (XT): the points that escape are those nearto, that is, are finite shifts of, the X s with s in the well-founded part of the tree .All such points are recursive. The points that abide are those near to X s with sin the ill-founded part, -those points again, individually, are recursive- and therecurrent points, which are exactly the points equivalent to the points xp placedat the end of each infinite path p. All recurrent points are non-HYPoThere are nominimal points, and all recurrent points are maximal.

Remark 6.2. Note that in these examples, if 'Y I>X s where s is in the ill-foundedpart of the tree, there is a sequence of points attacking 'Y, with each point of thesequence being recursive, but the sequence itself not even hyperarithmetic.

170 A.R.D.MATHIAS

7. Open problems

The present papers leave open the question whether there can be X, f and a withO(a, J) = WI: I would guess not, and that in all cases O(a,J) is at most the leastordinal not recursive in a and (a code of) f.

We collect here some thoughts directed to that question. Our examples are allin Baire space with the shift function unless otherwise stated.Proposition 7.0. There is a recursive a such that a A- {3 => {3 ~ HYP, butcontinuum many such (3 exist.Proof. Let C be a recursive tree with no HYP paths but with a continuum ofnon-HYP paths. Let Pi enumerate, monotonically, the odd primes, so Po =3. For0:f; SEC let 71"(s) = Ili<lh(s)Ps(i) , and let r(s) = (71"(s r k) 10 < k ~ lh(s)) . Sor( s) is a sequence as long as s of strictly increasing odd numbers, each dividingthe next with quotient an odd prime. Let r(0) = 0.

Let D = {r(s) Is E C}: then D also defines a closed set, and D has no HYPpaths, for recursive in any path through D is a path through C. Let a intersperseoccurrences of sED with occurrences of even numbers, such that no even numberoccurs more than once. Suppose that a A- {3. Then no even number can occur in {3,so each initial segment of {3 is a segment of something in D. So as in LD 1.13, {3 willbe a finite shift of some path through D. {3 itself will attack nothing. Finally, everypath through D, of which there are continuum many, is attacked by a . -j (7.0)

Corollary 7.1. There is a recursive a and a non-HYP"f such that a A- "f butthere is no {3 with a A- (3 A- "f.Proof. Let a be as in the previous example, and 'Y anything attacked by a. Then"f is not HYP, and there can be no f3 with a A- f3 A- "f, because anything attackedby a attacks nothing. -j (7.1)

Remark 7.2. That corollary shows the impossibility of proving that Aoo = AOby imitating the Lorenzen-Kreisel proof that Coo = Co. At the end of this sectionwe shall show that their ideas yield some information in this context.Problem 7.3. In that corollary, AI(a,s) is empty. Is there an example where Alis non-empty, and non-trivial, and A2 is empty?

An answer to that problem might perhaps be found by adapting the followinginstance of a case where A I is not closed. We work again in Baire space N withthe shift function s. Let Pi enumerate the primes, so Po = 2, PI = 3, P2 = 5, . .. j

let qi = P2i+I · Let e, = qi+1, an even number. For i, kin w, let 71"i,k = ~i+4)k+I ,

so these numbers are distinct powers of distinct odd primes> 5.We define a point z, points Yi, Xi for each i < w, and a point a, all in N, thus:

z =df qo ,qI, q2, q3, .. · ;

Yi =df qO,qI, · .. ,qi,ei,ei,ei, ' " j

Xi =df (71"i,O) ~ (Yi r ni,O) ~ (71"i,I) ~ (Yi r ni,d"" .. . ,

where the ni,k are chosen with i + 2 ~ ni,O < ni,I < ni,2 < ." j anda =df (xo r mo)~{5)~(XI rmd~(52)~(xo rm2)~ .. . ,

where the m's are strictly increasing and each Xi is visited infinitely often.

Proposition 7.4. z is not in Al(a,s) but is in its closure.

Proof. Evidently a r\. Xi rv Yi for each i, so that each Yi is in A l, and lim, Yi = z.

Suppose that a rv b r\. z: we shall derive a contradiction and thereby prove theproposition.

Note that ej occurs in Xk iff k = j, and that qi has exactly one occurrence ins» r nk,l if k ~ i and none if k < i , so that qi occurs in Xk iff k ~ i, Further, nopower of 5 can occur in b as each only occurs once in a, and so each finite segmentu C b is a segment of some Xk.

Each qi occurs infinitely often in b as b r\. z. Let u be a segment of b of lengthat least 2 that both begins and ends with qi. Then u C Xk for some k ~ ij the twooccurrences of qi must come from different initial segments of Yk and so betweenthem is an occurrence of ek.

So for some k, ek occurs in bj qk+l occurs in b, as all qi do; so some segment v

of b contains occurrences of both ek and qk+l' But no such v can be a segment ofany Xl, since if ek occurs in Xl then k = l; and if qk+l occurs in Xl, k + 1 ~ i.

-j (7.4)

7.5. Hausdorff's criterion for a set to be ~g, mentioned in Remark 1.5, enablesus to find cases where the set of recurrent points is not Eg. For a subset H of anappropriate topological space Hausdorff defines Hp to be ci(H) "H, and H", tobe Hpp • It is easily checked that H ;2 H",. Hausdorff proves in §27 of his bookMengenlehre that H is simultaneously F(1 and Go if and only if the (possiblytransfinite) sequence H, H"" H",,,,, ... , nn<w H",n, ... eventually reaches theempty set . In particular if H = H", :f:. 0, H cannot be ~g. Thus we obtain thefollowing:

Proposition 7.6. In the case of the shift function, both B = {,8 I ,8 r\. ,8} andB e = {,8 I ,8 rv £} are rrg, but neither is Eg.Proof. We prove that for H = B or Be, H = H1/>. Unravelling Hausdorff'sdefinition, we see that it suffices to show that each point in H is a limit of pointsin tr;H =Be: given ,8 r\. e, let 'Yn rn = ,8n, let the rest of 'Yn be something not attackings: possible provided {'Y I 'Y rv s] is not the whole space. Then let 8~ rm = 'Yn r mand let the rest of 8~ be something attacking e, e.g. {3, or, even better, ,8 rw" m.H = B: given,8 rv ,8, let 'Yn r n = {3n, let the rest of "[n. be something not attacking'Yn -note the self-reference. Then let 8~ rm = 'Yn r m and let the rest of 8~ besomething attacking 8~, e.g. some Cohen real. -j (7.6)

If we want to establish similar results for {,8 Ia rv ,8 r\. ,8} and {{3 Ia r\. ,8 rv'Y}, we have to build 'Yn, 8~ according to the following matrices.

[: =~ : :], [: =~ : ~]a r\. ,8 r\. e a r\. ,8 rv ,8

172 A.R.D.MATHIAS

But note that the constructions of Long Delays easily provide examples ofpoints a, e where these sets are countable, finite or even empty. Hence they mightbe ~g . So all we may hope to do here is give examples where they are not: but forsuch, any a with orbit dense in the space will do.

Problem 7.7. Where do the sets

{a Iwf(a) n B is Fu } and {a Iwf(a) n BE is Fu }

lie in the projective hierarchy?

Problem 7.8. Find a case where {a Ia minimal} is a complete IIt(f) set .

Example 7.9. Consider the space Q of sequences of length W of rationals. Foreach real A let Q<,\ be the subspace of sequences of rationals < A, and for Arational let Q~'\ be the subspace of sequences of rationals ~ A. Let p,\ be a pointof Q<,\ with orbit under the shift function 5 dense in Q<'\ , and for A rational letq,\ be a point of Q~'\ with orbit under 5 dense in Q~'\ , Then for reals A, J-L, andrational "1 with A < "1 < J-L,

while P,\ tA PT/ tA qT/ tA Pw Mapping to Baire space by a homoeomorphism preserving 5, we find there a set of points, each recurrent under 5, that is strictly linearlyordered by rv and is then order-isomorphic to the real line with every rationalpoint doubled.

Problem 7.10. Can we use the Gandy-Harrington topology to discuss minimalsets?

Problem 7.11. Is there an example where Al is strictly ~t ?

Theorem 7.12. Let a be the starting point, f the function. f is continuous andso is coded by a real which we also denote by f . Let 0 = 0(1, a, x) be the leastordinal not recursive in the triple (I, a, x) . Then

Proof. Towards the non-trivial direction, suppose that Vv < 0 x E AV.We consider trees T ~ S , the set of sequences of primes we have considered

before, with a top point w.An x-frame on such a tree is a function attaching to each node s a point Ys,

the top node must get point x, all other nodes must get a point y with a "" y rv x,and the attachment must be such that for s -< t, Ys "" Yt . So a frame with toppoint x provides evidence that x survives at least as long as the rank of the treesupporting the frame . We can construct frames by using the axiom of choice and

Lemma 7.13 (The Richness Lemma). Let x E AT/ and let ( < "1. Then 3y EA( with y rv x.

Proof. By induction on 1J. The lemma is vacuous for 1J =0; the induction is easyfor 1J a limit. For 1J =~ + 1, take two cases, ( < ~ and ( =~. -l (7.13)

We allude to the triple (/, a, x) as the data, and call something data-recursiveif it is recursive in the data.

Proof of 7.12 continued. Now the collection of trees coded by a data-recursiverelation on W for which there exists an x-frame is Et iL. a, x), and contains all datarecursive well-founded trees, by our assumption on x, using the Richness Lemma toconstruct the requisite frames. Hence there is an ill-founded tree in the collection:we shall call the frame on it a pseudo-frame to emphasize its ill-founded character.Any descending infinite path through the tree will therefore yield a sequence ofpoints x = Yo, YI , . . . such that for each i, a 0. Yi+l 0. Yi and YI 0. x, provingthat x E Aoo. -l (7.12)

Corollary 7.14. For any f, a, x, if x E A~ "A~+l then ~ is recursive in a, x, f .

Corollary 7.15. Suppose that O(a,J) = WI . Then U{wr IbEE} = WI . Hence Eis in this case a complete III set.

Similarly,

Proposition 7.16. Let a and f be recursive. Let 6 be the least non-recursive ordinal. If for each recursive u AII(a) is non-empty, then Aoo is non-empty.

Proof. For each e coding a recursive ordinal u there is a b and a v-frame for b.Hence there is a pseudo-frame for some b. -l (7.16)

Finally, we recast the above argument in terms of non-standard models, inanalogy to one approach to the Kreisel-Lorenzen result.

Proposition 7.17. Let x be in an ill-founded w-model N containing a and fwith an ill-founded ordinal c such that N l= x f AC and N l= The llichnessLemma. Then x E AOO .

Proof. Externally to N choose a descending sequence c, of ordinals of N startingfrom CO =c. Set Yo =x , and repeatedly apply the richness lemma to pick Yi+l ENsuch that N l= a 0. Yi+l 0. Yi and N l= YifAc;. Then this sequence is genuinely adescending attacking sequence and so each Yi is in Aoo. -l (7.17)

References

1. K.J .Barwise, Admissible Sets and Structures, Perspectives in Mathematical Logic,Springer Verlag, Berlin - Heidelberg - New York, 1975.

2. K.J .Barwise, R.O.Gandy and Y.N.Moschovakis, The next admissible set, Journalof Symbolic Logic 36 (1971) 108-120.

3. J . Harrison, Recursive pseudo-well-orderings, 1hmsactions of the American Mathematical Society 131 (1968) 526-543 . For a review, see [10] .

4. F. Hausdorff, Grundzuge der Mengenlehre, W. de Gruyter, Leipzig, 1914. (availablein an English translation)

5. G. Kreisel, Analysis of Cantor-Bendixson theorem by means of the analytichierarchy, Bulletin de l'Academie Polonaise des Sciences, Serle des sciencesmatMmatiques, astronomiques et physiques, 7 (1959) 621-626 .

6. R. Mansfield and G. Weitkamp, Recursive Aspects of Descriptive Set Theory, OxfordLogic Guides, # 11, Oxford University Press, 1985.

174 A.R.D.MATHIAS

7. A.R.D .Mathias, An application of descriptive set theory to a problem in dynamics, CRM Preprint mim. 308, Octubre 1995 (See alsohttp://www.dpmms.cam.ac.uk/ ardm/appdyn97).

8. A.R.D .Mathias, Long delays in dynamics, CRM Preprint mim, 334, Maig 1996 (Seealso http://www.dpmms.cam.ac.uk/ ardm/delays97) .

9. Y.N.Moschovakis, Descriptive Set Theory, North Holland , Amsterdam - New York- Oxford, 1980.

10. Y.N.Moschovakis, Review of [3], Journal of Symbolic Logic 31 (1972) 197-8.11. Y.N.Moschovakis, Review of [5], Journal of Symbolic Logic 35 (1970) 334.12. H. Rogers, Jr., Theory of recursive junctions and effective computability, McGraw

Hill, New York, 1967.13. J . Shoenfield, Mathematical Logic, Addison-Wesley, Reading, Massachusetts, 1967.

A TREE-ARROWING GRAPH

S. SHELAH1

Hebrew University,Jerusalem, Israel.

AND

E. C. MILNER2

University of Calgary,Calgary, Canada.

Dedicated to the memory of Eric Milner

Abstract. We answer a variant of a question of ROdI and Voigt by showing that,for a given infinite cardinal A, there is a graph G of cardinality K, = (2.\)+ suchthat for any colouring of the edges of G with A colours, there is an induced copyof the x-tree in G in the set theoretic sense with all edges having the same colour.

Keywords: Partition relation, graph , tree, cardinal number, stationary set, normal filter.AMS Subject Classification (1991): 03, 04

1. Introduction

9 = (V, E) is a graph with vertex set V and edge set E , where E ~ [V]2. The graph1l = (W, F) is a subgraph of 9 if W ~ V and F ~ E, it is an induced subgraph ifF =En [W]2. If A is a cardinal, the partition relation

9 -t (1l)L (1)

means that if c : E -t A is any colouring of the edges of 9 with A colours, thenthere is an induced copy of 'H in 9 in which all the edges have the same colour.There is a related notion 9 -t (1l)i, for vertex colourings of graphs. However,there is an essential difference since, for any given graph H and any A, there is

lpaper Sh 578 in Shelah 's publication list. Research supported by "The Israel ScienceFoundation" administered by The Israel Academy of Sciences and Humanities.

2Research supported by NSERC grant #69-0982.

175


176 S. SHELAH AND E. C. MILNER

some 9 such that 9 -t (1l)1 holds. This is not true for edge-colourings; Hajnal andKomjath [2] proved the consistency of a negative answer, and Shelah [5] provedthat a positive answer is also consistent. It is therefore of some interest to haveinstances of graphs H such that (1) holds for some g, and then, of course, one canask for the smallest such g.

Rodl and Voigt [4] (see also [3]) proved a result of this kind by showing thatfor any infinite cardinal Aand a suitably large n, there is a graph g" of cardinality/\, such that

(2)

holds, where", is the tree in which every vertex has degree /\, (see below) . Moreprecisely, 'suitably large' means that the ordinary partition relation

holds so that, by [1], /\, ~ (22")+; in fact, they showed in this case that the ubiquitous shift-graph on /\, works. Rodl and Voigt [4] then asked , what is the smallestcardinal x such that (2) holds? It is easily seen that (2) is false if /\, ~ 2'\, and theyconjectured that it holds (for some suitable graph g,,) if /\, = (2,\)+ . In this paperwe prove that (2) holds with", replaced by 7(/\'), a related graph which we callthe transitive s-tree defined in the next section.

2. Preliminaries

For an infinite cardinal x we denote by <w /\, the set of all increasing finite sequencesof ordinals in n: The length of an element s = (so, .. . , Sn- l ) E <w /\, is denoted byIn(s) =n. Also, we define

{- I

max(s) =Stn(s)-l

if s = (), the empty sequence,iUn(s) > O.

If s = (so, ... ,Sn-l) and t = (to, ... ,tm-l) are two elements of <w«, we writes <l t to denote the fact that s is a proper initial segment of t, that is n < mand Si = ti for i < n, and in this case we write s = tin. We also write s = t; ifm =n + 1 and s <l t, If s, t are distinct and <l-incomparable we write s .L t. Then-iree of height w is the graph", on <w /\, with edge set

E" = {{s,t} : s,t E <w/\,/l.s = t.}.

We shall also consider a related graph, the transitive n-ttee of height w, which isthe graph 7(/\') on <w /\, with edge set

Fit = {{s,t}: s,t E <w/\,/I. s <l t}.

We shall prove the following theorem.

A TREE-ARROWING GRAPH 177

Theorem 2.1. Let>. be an infinite cardinal, and let K. = (2)')+. Then there is agraph GK of cardinality K. such that

GK -t (nl,

where T is T(K.).Remark. Instead of K. = (2)')+ , it is enough that K. be any regular cardinal suchthat 101>' < K. holds for all a < K.. The same proof works.

The construction of a suitable QK depends upon the following (slightly weakerversion of a) theorem of Shelah [7J (or more [8, 3.5]):

(e) Let>. be an infinite cardinal, K. = (2)')+,8 = {a < K.: cf(a) = >.+}. Thenthere are a sequence C = (Co: 8 E 8) and a sequence h* = (h5 : 8 E 8) suchthat Co is a club in 8 having order type >.+, h5 : Co -t 2 and such that, forany club K in K., there is a stationary subset BK of 8 n K such that for each8 E BK and each i < 2, min(Co) E K and the set

DK(8,i) = {a E Co n K: h5(a) = i /\ min (Co \ (a + 1)) E K}

is cofinal in 8.

Remarks. 1. The result is also true if 2, the range of each h5, is replaced by >'ialso, if K. = >.++ , we can also require that D K (8, i) be a stationary subset of 8 foreach8 E BK and i < >. (see [8]).

£. If 2>' > >'+ , then the following stronger assertion is true (see Shelah [6]):(..)There is a sequence C = (Co : 8 E 8) such that Co is a club in 8 having ordertype >.+ and, for any club K in K. and any stationary subset 8' ~ 8, there is astationary subset BK ~ 8' n K such that Co ~ K for each 8 E BK. Using thisresult instead of (e) , the proof of Theorem £.1 for the case when 2>' > >.+ may beslightly simplified.

We will prove that Theorem 2.1 holds with the graph GK = (K. ,c), where

e = {{a,,8} :,8 E S /\ min(CI3) < a <,8/\ hp(sup(a n CI3)) = O},

and the CI3 and hpare as described in (e).

3. The case T =T(K.)

We prove the result for the case of the transitive tree T(K.).

Proof: Let c : £ -t >. be any >.-colouring of the edges of GK • For each ( E >.consider the following two-person game QC; . The game has w moves. At the n-thstage the first player PI chooses ordinals an, ,8n, and then the second player P2

chooses two ordinals "In, 8n so that

8m < an (m < n) .

(3)

(4)

The player P2 is declared the winner in a play of the game if he succeeds in choosingthe "In so that

and

c(hm,"In}) =( (m < n < w), (5)

(6)

(As usual, (a,{3) denotes the open interval {~ : a < ~ < {3} and [a,{3] is thecorresponding closed interval.)

The proof of the theorem depends upon the following two facts:Fact A: For some ( < A, P2 has a winning strategy for the game Q(.

Fact B: If P2 can win Q(, then the graph G/f, contains an induced copy of T(K,)with all edges coloured (.

Proof of Fact B. We assume that ( < A and that the second player P2 has awinning strategy u( for the game Q(. We shall define ordinals as, (3s, "Is, 8s for s avertex of T(K,) so that the following conditions are satisfied:

(a) For each s the sequence

((asli' (3sli ,"Isli, 8sli) : i < In(s))

consists of the first 2in(s) moves in a proper play of the game Q( in which P2 usesthe winning strategy ac-

(b) "Is # "It if s # t .

(c) If s.l t, then hs,"It} ¢ c.Since (5) holds, these conditions imply that the map s I-t "Is is an embedding

of the tree T(K,) into the graph G/f, and all the edges of the image have colour (.

In fact, we shall choose the as, (3s, "Is, 8s so that (a) holds and so that thefollowing condition is satisfied:

(d) For any vertices s, t of T(K,), if s .1 t, then

EITHER (i) ["Is,8s] C Ui~ln(t)(atli,{3tli)'

OR (ii) ["It,8t] C Ui9n(s) (asji, (3sli) .

The conditions (a) and (d), and the fact that P2 is using the winning strategy u(,ensure that (b) and (c) also hold.

We define as, {3s,"Is, 8s by induction on max(s). Let 0.0 = 0, (3o = 1, and thenlet ho, 80) be P2's response in the game Q( using his winning strategy u(. Now leto~ ~ < K" and suppose that we have suitably defined as, (3s, "Is, 8s for all verticess of T(K,) such that max(s) < ~. We need to define these when max(s) = ~.

Let (ti : i < O(~)} be an enumeration of all the nodes s of7(K) with max(s) = ~.

Then 1 ~ O(~) ~ 2A < K. Now inductively choose the at" (3t" 'Yt" 6t, for i < O(~)so that

at; = 6(t;j. + 1,

and if i =0, {3t , =aio + 1 and if i > 0

(3t , =sup{6s + 2: max(s) < ~ or s = tj for some j < i}.

The corresponding pairs (-Yt" 6tJ are determined by the strategy ac- With thesechoices it is easily seen that (a) continues to hold; we have to check that (d) alsoholds when s 1- t and max(s) = ~ or max(t) =~.

If max(s) = max(t) =~, then s = ti and t = tj, where say i < j. Then

at = 6t• + 1 < {3s < 'Ys < 6s < e;and so (d)(i) holds.

Suppose max(s) < ~ = max(t) . Then by the induction hypothesis , either (i) or(ii) of (d) holds when we replace t by t •. Suppose first that (d)(i) holds. Then forsome m ~ In( t.) we have that

at.lm < 'Ys < s, < (3t.lm '

It follows that (d)(i) also holds for s and t since tim = t.lm. Now suppose that(d)(ii) holds so that, for some m ~ In(s),

aslm < 'Yt. < s., < {3slm '

Then, by the definitions of at and {3t, it follows that

at = 6t• + 1 ~ {3s < 'Ys < s, < (3t,

so that again (d)(i) holds for s and t. Similarly, if max(t) < ~ = max(s) . 0

Proof of Fact A. We have to show that P2 wins the game Q< for some ( < A.Suppose for a contradiction that this is false. Since the games are open and hencedetermined, it follows that PI has a winning strategy, say T<, for the game Q< forevery « A.

For convenience we write c({a, (3}) = -1 if {a, {3} ¢ £, so that c is definedon all pairs {a, {3} E (KJ2 . For each bounded subset X ~ K define an equivalencerelation ex on S \ (sup(X) + 1) so that {3 ex 'Y holds if and only if

(i) (3, 'Y E S and sup(X) < {3, 'Y < Kj

(ii) c({a , (3}) = c({a, 'Y}) for all a EX;

(iii) X n C{3 = X n C'Y' (iv) for a E X, a ~ min(C{3) ¢} a ~ min(C'Y),tp(a n C(3) = tp(a n C'Y) and hp(sup(an C(3)) = h;(sup(a n C'Y)) (fora> min(C{3)) .

Note that the equivalence relation ex has at most (>.+) IXI ~ 2,xlxl classes. Also,if y ~ X, then (3 ex 'Y ::} (3 ev 'Y.

Since /'i, = (2,x)+, there is a continuous increasing sequence of ordinals(PfJ : 1] < /'i,) in /'i, such that the following two conditions hold:

(0) If X ~ PfJ' IXI ~ >. and PfJ < (3 < /'i" then there is some 'Y E (PfJ,PfJ+dsuch that (3ex'Y

(00) PfJ is closed under 7"( for all ( < >.. In other words , if at the n-th stage of aplay in the game ~h, player P2 chooses 'Yn < 8n < PfJ' then PI'S responseusing 7"( is to choose O:n+!, (3n+! so that 8n < O:n+! < (3n+! < PfJ'

Since K = {PfJ : 1] < /'i,} is a club in /'i" there is some 8 E S such that min( Co) EK and, for e E {O, I},

AE = {o: E Co nK: h6(o:) =e A min (Co \(0:+ 1)) E K}

is an unbounded subset of 8. Let Co = {i17 : a < >.+}, where io ..

(*k If X ~ 8, IXI ~ >., then there are a < >.+ and 'Y such that (a) sup(X) <i17 < 'Y . there is a counter-example

X(. Let X =U{X( : ( < >.}. Then X ~ 8 and IXI ~ x and so, for some 0: E Ao,sup(X) < 0: < 8. There are 1] < /'i, and a < >.+ such that 0: = PfJ = i 17 , andtherefore, by the choice of PfJ+!' there is 'Y such that PfJ < 'Y < PfJ+! and 'Yex8.Since 0: = i 17 E Ao, i 17+! = min(Co\ (0 + 1)) E K. So PfJ+! ::;; i 17+! . Therefore,sup (Co n 'Y) = i 17 , and since 0: = i 17 E Ao, we have that h6(sup(Co n 'Y)) = O.Therefore, b, 8} is an edge of G and there is some ( E >. such that c('Y, 8) = (.But this contradicts the choice of X( ~ X, and hence (*k holds for some ( < >..

By induction on n < w we now choose ordinals O:n, (3n, 'Yn , 8n in 8 and a(n) < >.+so that the following conditions are satisfied:

A: ((O:m, (3m, "[m» 8m) : m ~ n) is an intial segment of a play in the game 9( inwhich PI uses the winning strategy 7"(.

B: 0:0,(30 < min(Co).

C: 'Yn =minb : 'Y > i 17(2n) A 'YeXn 8 Ach,8) =0, where

D: 8n = i 17(2n+I)'

E: For n > 0, [O:n,(3n] ~ (8n- l , i17{2n- I)+d .F: i 17{n ) belongs to Ao or Al according as n is even or odd and a(n) +1 < a(n+1).

We have to prove that it is possible to choose the O:n etc., so that these conditions are satisfied. Clearly (B) holds since, by (00), the first moves by PI using thestategy 7"( are 0:0 < (30 < Po and Po ~ min(Co) E K. By (*)<; , there are a(O) < >.+

and 'Y such that iO'(o) E Ao, iO'(o) < 'Y < iO'(O)+l' 'Yexo 0, where Xo = {ao,.8o}and c{"o) = (; let 'Yo be the least such 'Y. Now let u(l) > u(O) + 1 be minimalso that iO'(l) E A1 , and put 00 = iO'(l) ' Now suppose that n > 0 and that theam, .8m, "[m ,Om, u(2m) and u(2m + 1) have been suitably defined for all m < n .Let p E K be minimal such that p > On-l . P1 chooses an,.8n using the strategy T( so that On-l < an < .8n < p. Since On-l = iO'(2n-l) E A1, it followsthat iO'(2n-l)+1 E K and hence p ~ iO'(2n-l)H' Now by (*k, there are u(2n)and 'Y so that iO'(2n) E Ao, iO'(2n) < 'Y < iO'(2n)H' "t ex; 0 (where Xn is as described in (C)), and c('Y,o) = (; let 'Yn be the least such 'Y. Note that, sinceiO'(2n) E Ao, iO'(2n)H = min(Co \ (iO'(2n) + 1)) E K . Finally, choose a minimal ordinal u(2n+ 1) > u(2n) +1 so that On = iO'(2nH) E A1. This completes the definitionof the an etc ., so that (A)-(F) hold.

By (C) it follows that c{,n,o) = ( for all n < w, and hence c{,m, 'Yn) = (holds for all m < n < w since 'Ym E X n and "t« ex; O. There is no edge of GK,from 0 to (ao,.8o) since .80 < min(Co)' Since 'YneXn 0 and .80 E Xn, it followsthat .80 < min(C-yJ also, and so there is no edge from 'Yn to (ao,.8o) either.By the construction, for 0 < m < w, iO'(2m-l) < am < .8m < iO'(2m-l)+1l andhence Co n (am,.8m) = 0. Therefore, for any ~ E (am,.8m), hHsup(~ nCo)) =hHiO'(2m-l») = 1 by (F), and so there is no edge of G from 0 to (am, .8m) . If0< m < n < w, then 'Yn ex; 0 and therefore,

tp(am n C-Yn) =tp(am nCo) =tP(.8m nCo) =tP(.8m n C-Yn) '

Therefore, for any ~ E (am, .8m), it follows that

and so there are no edges of G from 'Yn to (am,.8m) either.

Thus we have produced a play in the game Q( in which P1 uses the strategyT( but the second player P2 wins! This contradicts the assumption that u( is awinning strategy for the first player, and completes the proof. 0

References

1. P. Erdos and R. Rado, A partition calculus in set theory , Bull. Amer. Math. Soc.62 (1956) 427-489.

2. A. Hajnal and P. Komjath, Embedding graphs into colored graphs, TI-ans. Amer.Math. Soc. 307 (1988), 395-409 ; Corrigendum: 332 (1992), 475.

3. P. Komjath and E.C. Milner, On a conjecture of Rodl and Voigt. J . Combin. Theory,Ser. B 61 (1994), 199-209.

4. V. Rodl and B. Voigt, Monochromatic trees with respect to edge partitions, J.Gombin. Theory Ser. B 58 (1993), 291-298.

5. Saharon Shelah [Sh: 289], Consistency of positive partition theorems for graphs andmodels, in: Set theory and its applications (Toronto, ON, 1987), Lecture Notes inMathematics 1401, (J . Steprans and S. Watson, eds.), Springer, Berlin-New York,(1989) 167-193.


6. Saharon Shelah ISh: 365],There are Jonsson algebras in many inaccessible cardinals,in: Cardinal Arithmetic, Oxford Logic Guides 29 chapter III, Oxford UniversityPress, 1994.

7. Saharon Shelah ISh: 413], More Jonsson Algebras and Colourings, Archive forMathematical Logic, to appear.

8. Saharon Shelah ISh: 572], Colouring and N2-cc not productive, Annals of Pure andApplied Logic, 84 (1997), 153-174..

A HOLLOW SHELL: COVERING LEMMAS WITHOUT A CORE

W. J. MITCHELL!Department of MathematicsUniversity of FloridaGainesville, FL 32611USA

Abstract. This paper is an attempt to apply the proof of the covering lemma insituations where the usual statement of the coveringlemma is meaningless becauseno core model exists. The main result is the following theorem:

Theorem. Suppose L[E] is a minimal class model for a Woodin cardinal and thecovering lemma over L[E] fails at a cardinal fJ such that that there is no iterableL[E]-ultrafilter on any cardinal TJ < fJ of L[E] . Then every cardinal v in the intervalfJ < u < sup(domain(E)) is the limit of a closed and unbounded set of weaklycompact cardinals of L[E] .

Many different forms of the covering lemma have been published since its firstappearance in Jensen 's handwritten notes "Marginalia on a Theorem of Silver"(see [2]). These variations share both a basic similarity in their statement-thatif the core model K exists then the core model is close to V -and a commonbasic method of proof. The meaning of the word "close" has varied, becomingweaker and more complicated as the hypothesis has become weaker, but all formsof the covering lemma have as their most important consequencethe weak coveringlemma: If TJ is a singular cardinal in V then TJ+ (K) = TJ+.

Except in quite recent work, the condition for the core model to exist has beensimply that there was there did not exist any inner model containing cardinalstoo large for the technology of the core model in question. This holds true forthe Jensen's original work with L [6, 2], for Dodd and Jensen's work below ot[3] , and for Mitchell's work with sequences of measures [10, 11]. It began to breakdown, however, in the work of Steel [21, 20, 19] and of Steel and Schimmerling[18] , who have constructed core models up to a strong limit of Woodin cardinals.These results require a new hypothesis in addition to the usual assumption thatthere is no model with cardinals which are too large to handle: the construction ofthe core model takes place entirely below some large cardinal n. It is now known

IThis work was partially supported by NSF grant number DMS-9306286

183

184 W. J. MITCHELL!

that n can be much weaker than a measurable cardinal, but some strength is stillneeded. This extra assumption leaves gaps in the coverage of these theorems: forexample, if one assumes that there is a model with a Woodin cardinal, but nosharp for such a model, then there is no core model; yet there exist core modelscontaining far more than a single Woodin cardinal .

Woodin cardinals present a more serious problem: Woodin, following ideas ofForeman, Magidor and Shelah, has used stationary tower forcing [22, 23] to showthat if M is any model with a Woodin cardinal ~ and TJ < ~ is singular in M thenthere is a set generic extension M[G) of M such that in M[G] the cardinal TJ ispreserved while TJ+(M) is collapsed. This implies that there cannot be a core modelfor a Woodin cardinal which simultaneously satisfies the weak coveringlemma andis definable by a formula which is absolute for set forcing.

This paper is an attempt to make a start at understanding what can be madeof the covering lemma in such a setting. One possible precedent is the DoddJensen covering lemma for L[Jt]. Even before Jensen's original covering lemmawas known, Prikry [15] had shown how forcing can be used to make a measurablecardinal singular, so that the statement of the coveringlemma for L is consistentlyfalse for L[Jt] . Dodd and Jensen showed, however, that if ot does not exist thenPrikry forcing is the only way in which that statement can fail.

We would like to similarly classify the possible ways in which the weak coveringlemma can fail at a Woodin cardinal . It is unlikely that there is a straightforwardstatement such as that for L[Jt], but it is natural to conjecture that any such failuremust resemble, at least locally, one of the two known forcing extensions which canbe used to violate the weak covering lemma. These two forcing extensions arethe stationary tower forcing refered to previously and the remarkable, and so farunpublished, "all sets are generic" forcing, also due to Woodin. The current paperis intended to be a start on this program, but it is no more than a start. The resultsin this paper are deficient in at least three respects: (i) they do not deal usefullywith overlapping extenders, and hence only deal with a failure of the weak coveringlemma occurring below the least potentially measurable cardinal, (ii) they do notreally deal with Woodin cardinals and hence are limited to the minimal model forWoodin cardinal, and (iii) their conclusion is far weaker than the known notions offorcing would suggest, even when the first two limitations do not apply. The firstpart of our title was chosen as much for its idiomatic meaning as for its physicalimage. It is hoped that future research will enable us to overcome-or at least tobetter understand-the limitations of these results.

We would like the thank the referee for a number of helpful suggestions andcorrections to an earlier version of this article.

1. STATEMENT OF RESULTS

We will be dealing with models of the form L[£], where E is a good sequence ofextenders in the sense of [13]. These models are not, in general, fully iterable : theremay be iteration trees 7 on the model L[£] such that no cofinal branch of 7 has a

A HOLLOW SHELL 185

well founded limit . The models will be iterable, however, in the following weakersense due to Woodin.

Definition 1. - If T is an iteration tree then we write £r for the sequence ofextenders stabilized by I. That is, £r = U{£o fpo : a < len(T) }, where £0is the extender sequence of the oth model of T and Po is the index of theextender used at the oth stage.We say that M is O-iterable if it is well founded, and M is n + l-iterable if,for every tree T on M, either T has a cofinal branch with a n-iterable limitor else L[£r] satisfies that there is a Woodin cardinal.We say that M is iterable if it is n-iterable for each n < w.A model M can be iterated to a Woodin cardinal if there is an iteration treeT on M such that L[£r] contains a Woodin cardinal.

If the tree T comes from the comparison of L[£] with a second model then £ris the sequence of extenders which are matched by the comparison.

It should be noted that we are interested in the iterability of L[£] in V, notin the model L[£] itself. The trees T', as well as the required branches, will not ingeneral be members of L[£]. The basic result on iterability is the following lemmaof Martin and Steel [8] :

Lemma 2. If a tree T on the model M has two distinct branches band c suchthat each of the limits Mb and Me has a well founded part of length at least 'Y,then L-y[£r] satisfies that len(£r) is Woodin.

Hence, if M is an iterable model and T is a tree on M with a branch b suchthat Mb has a well founded part of length 'Y and L-y[£r] satisfies that len(£r) isnot Woodin, then Mb is fully well founded and b is the unique well founded branch«t:

If a model M is not iterable then it means that there is a finite sequence ofnormal trees, such that the first tree starts with M, the other trees start with thelast model of the one before, and the last tree neither has a well founded branchnor generates a Woodin cardinal. It will be convenient to consider such a sequenceof normal trees as a single, piecewise normal, tree.

We will call such at tree T witnessing the failure of iterability badly behaved,and we will say that T is badly behaved at 'Y if L-y[£r] satisfies that len(£r) is notWoodin. The following observation of Woodin is crucial:

Lemma 3. If T is a tree on M, then T is badly behaved at 'Y in V if and only ifit is badly behaved at'Y in V[G], where G is a Levy collapse of'Y to w.

Thus the statement that T is badly behaved at 'Y is absolute for well foundedmodels containing 'Y and I.

Sketch of proof. If T is not badly behaved at 'Y in V then either L-y[£r] satisfiesthat len(£(T)) is Woodin or else T has a branch b in V such that the limit Mbhas a well founded part of length at least 'Y. In either case it follows that T is wellbehaved in V[G] .

Now suppose that T is badly behaved at 'Y in V. If it is not badly behaved at'Y in V[G] then T must have a branch in V[G] which which has a limit with wellfounded part of length at least 'Y. This branch is is unique by Lemma 2 and it

186 W. J. MITCHELL!

follows by the homogeneity of the Levy collapse that this branch already exists inV. This proves the first paragraph.

The statement that "len(£r) is Woodin in L")'[£rJ" is absolute for any model.The statement "7 has a branch b such that the limit Mb has well founded part oflength "t" is equivalent to the statement that there is an infinite branch througha certain countable tree in V[G]. Putting these two observations together showsthat "7 is badly behaved at "t after a Levy collapse of "t to w" is absolute; andthe second paragraph of the lemma then follows from the first paragraph. 0

The following definition is taken from Kunen [7J except that we require that Ube normal :

Definition 4. AM-ultrafilter on a model M of set theory is a non-principalultrafilter U on some cardinal a of M such that if x E M n P(a) and IxlM = athen xnU E M, and such that if f : a -t a is in M with {v : f(v) < v} E U thenthere is ~ < a such that {v : f(v) =0 E U.

We call an L[£]-ultrafilter U iterable if ult(L[£], U) is wellfounded and iterable.

Unless stated otherwise, the word "branch" will always mean a cofinal branch.We are now ready to state the main theorem. The reader may wonder about

the fact we do not rule out the possibility that £~ exists or that K, is collapsed inV, but a moment's thought shows that that the theorem is true, though trivial, insuch a case.

Theorem 5 (Main Theorem). Let L[£] be an iterable model such that L")'[£]cannot be iterated to a Woodin cardinal for any ordinal"t < len(£) . Set K, = len(£)if L[£J can be iterated to a Woodin cardinal, and let K, = Ord otherwise. Finally,let 8 be a regular cardinal of L[£] such that there is no iterable L[£]-ultrafilter onany cardinal a ~ 6 of L[£], and suppose the weak covering lemma fails at 6-thatis, 6 is singular in V , and cf(6)W < 161, but (6+)L[t:) < 6+.

Then v+(L[t:)) < v+ for every ordinal v in the interval 6 < v < K,. Indeed forany cardinal a of V in the interval 6 < a < K, there is a closed and (if cf(a) > w)unbounded subset Co of a such that

1. If v E Co then v is weakly compact in L[£J, and in fact v -t (stationary)~.

2. If u, jl E Co then cf(v+L1t"J) = cf(jl+L[t"J).

3. If v E Co and cf(v+L1t"l) > w then v -t (stationary)~w. In particular, u isRamsey in L[£].

4. If v E Co is a cardinal then CII = Co n u,5. Let i: L[£] -t M be the iterated ultrapower defined by taking a single ultra

power by the order 0 measure on every ordinalu E Co which is measurable inL[£] . Then Co is a set of indiscemibles for M .

The last clause essentially says that the members of Co are indiscernibles inL[£] except for the fact that they may differ in their degree of measurability.

If there is no iterable model with a Woodin cardinal then the theorem wouldonly be of interest if the Steel core model fails to exist; in this case any iterablemodel L[£] will satisfy the hypothesis of the theorem. If there is a model with aWoodin cardinal then the theorem is only of interest if there is such a model which

A HOLLOW SHELL 187

also has the required minimality property. Before continuing we verify that suchmodels do in fact exist.

Proposition 6. If there is an iterable model with a Woodin cardinal then there isan iterable model L[c] in which len(c) is Woodin, but L[cfa] cannot be iterated toa Woodin cardinal for any a < len(c).

Proof. We follow the obvious process to obtain the required model. Let Mo= L[co]be any iterable model such that len(co) is Woodin. Now suppose that Mn =L[cn ]

has been defined and satisfies that len(cn) is Woodin. If Mn satisfies the conclusionof the proposition then we are done; otherwise let an < len(£n) be the least ordinala such that L[£n fa] can be iterated to a Woodin cardinal using a tree Tn, and letMn+! = L[£r..] be the model obtained by this iteration.

We need to show that this process eventually stops, so we assume for the sakeof a contradiction that it does not. The contradiction depends on the ordinals anand does not involve the Woodin cardinals, so we can simplify the constructionby replacing each of the trees Tn with Tn fVn where Vn is the least ordinal v suchthat len(£rn til) > an+!' Then none of the models have a Woodin cardinal, and wehave proved the following claim:

Claim A. If the process above can be continued for infinitely many steps, thenthere is a sequence (Tn : n < w) of trees such that if we write Mn ,lI = L[cn ,lI] forthe vth model of Tn then

1. Mo,o = L[£Mo fao] for some ordinal ao.2. For each n < w, Tn is a tree of successor length On + 1 on Mn,o, with last

model Mn,8n =L[Cn,8n ] '

9. For each n < w there is an ordinal an < len(£n,8J such that

4. None of the models Mn , lI contain a Woodin cardinal.

Call such a sequence a fiojita below ao on the model Mo = L[CMo]. We willprove the following claim:

Claim B. Suppose that M is an iterable model, that G is V -generic for the genericcollapse of some ordinal T to w, and that there is a fiojita below a on M in V[G].Then there is such a fiojita in M[G].

In the following, we will work in a collapse V[G] of a cardinal T large enoughthat any flojitas relevant to the proof exist in V[G], if they exist in any genericextension of V.

To see that Claim B implies the proposition, let Mo and ao be minimal in thesense that there is a flojita below ao on Mo in V[G]; but there is no flojita on anymouse M' E Mo and no flojita below any a < ao on Mo. Fix some flojita f onMo·

By Claim B, the minimality of Mo and ao can be expressed in Mo, and henceis preserved by the tree embeddings. It follows that if there are no drops on themain branch of 10 then the last model M O,90 of 10 does not have a flojita below

188 W. J. MITCHELL l

any a < i6?8o(aO) = len(£Mo•Bo ) . If, on the other hand, there is a drop in themain branch of 70 then let MO,II+! be the first model after the drop. There is noflojita on MO,II+l , and this fact will be preserved by the tree embeddings so thatthere is no flojita on MO,80 ' In either case, we have a contradiction to the fact that(In : 1 $ n < w) is a flojita below al on MO,80' and this contradiction proves theproposition.

Now we turn to the proof of Claim B.

Proof of Claim B. If there is a flojita in V[G], then there is a countable flojita. Let

'Y < wi lG1be large enough that there is a flojita such that all of the trees of theflojita have length less than 'Y and for each model M n , 1I of the flojita, L-y[£Mn

. ... ]

satisfies that len(£Mn. ... ) is not Woodin. By increasing T if necessary, we can assume

that 'Y is countable in M[G].Define in M[G] a tree T, the nodes of which are attempts to construct a flojita

as in the last paragraph such that all of the models in the trees of the flojita havea well founded part of length at at least 'Y. The existence of the flojita in V[G]ensures that T has an infinite branch, and by absoluteness of well foundedness itfollows that T has a branch in M[G]. Thus it is only necessary to verify that thisbranch generates a flojita on M. The only problematic part of this construction isthe verification that the models of the flojita are well founded, but by Lemma 2this follows from the iterability of M and the fact that all of the models have wellfounded part of length at least 'Y. 0

This completes the proof of Proposition 6. oBefore giving the proof of the main theorem, we state one further result. The

proof of the following proposition, which will be given after that of the maintheorem, depends on the fact that the sets Ca come from the branches of aniteration tree, and on the observation that the lack of measurable cardinals in theinterval (a, (3) means that the relevant section of the iteration tree is linear. Itfollows that if a < v < v' < {3 then CII = CII' n v unless T has a drop between vand v', and the proposition follows from the fact that there are only finitely manysuch drops.

Proposition 7. Suppose that L[£] is as in Theorem 5, that d < a < {3 < K, andthat there are no measurable cardinals v in the interval a < v < {3. Then thereare at most finitely many cardinals u of V in the interval a < v < (3 such thatcf(v+L[£I) > wand v is not Ramsey in L[£].

The exception for finitely many cardinals cannot be removed: we will show inProposition 9 that if d is any finite set of cardinals of L[£] which can be preservedin a stationary tower extension, then the members of d can be preserved whiletheir successors are given uncountable cofinality.

2. PROOF OF THE MAIN THEOREM

There are two parts to the proof of Theorem 5. The first part of the proof isessentially the classical proof of the covering lemma. There are some complications

A HOLLOW SHELL 189

due to the fact that L[E] need not be fully iterable, but the existence of overlappingextenders does not otherwise cause any significant problems. The avoidance of suchproblems depends heavily on the hypothesis that there are no L[E]-measures onordinals v 5 8, so any attempts to strengthen Theorem 5 are likely to require newideas.

As in the proof of the covering lemma, the first part of the proof leads toa mouse M which witnesses that 8 is singular. If we were proving the coveringlemma we would then compare the mouse M with L[E], and the outcome of thecomparison would show that M E L[E], concluding the proof of the theorem. Thisargument breaks down in the current situation because the trees involved mayreach a Woodin cardinal and have no well founded branches. In the second part ofthe proof we will look at these iteration trees, deriving the required sets Ca fromtheir branches.

Part one of the proof of the main theorem. We now begin the first part of theproof. We can assume without loss of generality that 8 is not measurable in L[E],since if the theorem fails for a model L[E] having a a measure U on 8 in L[E] thenthe theorem also fails for ult(L[E],U) . We will write 8$ for (8+)L[el . Let X be anelementary substructure of H(26)+ such that

1. E, 8 E X and X n 8$ is cofinal in 8$.2. IXI = (cf(8$))W < 83. wXCX.

Let rr: N ~ X -< V· be the inverse of the transitive collapse, and set W =rr-1(L[E]). We will compare L[E] with W, using iteration trees T on L[E] and Uon W, and we will continue the comparison until the models of the trees agree upto 8 = rr- 1 (8). We will, of course, have to verify that the comparison terminatessuccessfully, without reaching a stage at which one of the trees does not have anywell founded branch.

Let M v be the vth model of T.Claim A. None of the models M; for v > 0 are proper classes.

Proof. Let TJ = crit(rr) . Ifp(TJ)nL[E] eN then {x : TJ E rr(x)} is an L[E] ultrafilteron TJ < 8 which is countably complete and hence iterable, contrary to the hypothesis of the theorem. Thus the tree T drops immediately: M1 is the ultrapowerof a mouse MJ of L[E]. None of the extenders used in T can have critical pointa smaller than the projectum of MJ, since if they were then they would in factbe extenders on L[E], contradicting the assumption that there are no measurablecardinals in L[E] below 8. It follows that the tree is effectively on the mouse MJ,so that M; is a set for all u > O. 0

Claim B. The model W is not moved in the comparison, that is, U is trivial.

Proof. Suppose to the contrary that W is moved in the comparison. Since thereare no measures in W on cardinals TJ 5 8 = rr-1(8), the tree U must start bydropping to a mouse: that is, there is a mouse no in W \ M; where v is the leastordinal such that M; agrees with L[E] up to the projectum p of no. As in Claim A,

190 W. J. MITCHELL!

the tree U must effectively be a tree on no, since if an extender F in the tree U hadcritical point smaller than the projectum of no then the measure of the extender Fwould be a measure on W, which would imply that W has a measurable cardinalless than 6.

Now continue the comparison, using iteration trees U' and T' extending U and7, until (if possible) the final model of one is an initial segment of that of the other.Since each model M; in T' is an iterate of a mouse from L[£], the hypothesis ofthe theorem implies that L[£r] does not have a Woodin cardinal, and hence 7has a final model M",. Likewise no is iterable, since a tree witnessing a failure ofiterability could be mapped via 7r to a tree on L[£] witnessing that L[£] is notiterable. The tree U also has a final model n, since otherwise L[£u] would have aWoodin cardinal, contradicting the fact that £u = Gr .

It follows that one of the premice n and M I1 is an initial segment of the other,but this is impossible since each of the premice has a definable subset which is notdefinable in the other. This contradiction completes the proof of the claim. 0

In particular the iteration succeeds, so that 7 has a final model M6 such thatthe extender sequences of M6 and L[£] agree up to 6ffJ =7r-1(offJ ). As in the proofof the covering lemma, let M be the least (possibly proper) initial segment ofM6. such that there is a bounded subset of J which is definable in M but is notin W. Now set M = ult(M,7l",offJ), where the ultrapower is taken by using theleast En code of M which has its E1 projectum less that J (or, equivalently, theultrapower uses functions which are En definable in M. Then M is (if iterable)also a mouse and the following diagram commutes, where hM and hM are theEn+1 Skolem functions:

M' ". .

M--t

hMj hM j6ffJ - • tSED".-". offJ

) .The proof of the next claim is essentially due to Woodin, using basic ideas of

Martin and Steel.

Claim C. The model M is iterable.

Proof. Suppose to the contrary that S is a tree on M which witnesses that M isnot iterable. This means that there is no well founded branch through S and thatlen(£s) is not Woodin in L[£s]. Let 'Y be the least ordinal such that a =len(£s)is not Woodin in L-y[£s]. Take TJ large enough that everything which has beenmentioned is in VI1 , let Y -< VI1 be a countable elementary substructure containingeverything which has been mentioned, and let k: Y ~ Y be the inverse of thetransitive collapse. Since Y is countable and W X C X, the tree k-1(8) can beembedded into a tree S' on M; as in Lemma 3.13 of [14]. The model M; isiterable, since it is obtained by a tree iteration of L[£], and as in Claim B itcannot be iterated to a Woodin cardinal . Thus S' has a well founded branch, andit follows that the corresponding branch of k- 1(8) also has a well founded limit.

A HOLLOW SHELL 191

It will be enough to show that this branch is a member of Y, since by theelementarity of k this implies that Y contains a well founded branch of S, contradicting the choice of S . Let G be generic over Y for the Levy collapse of k-1b),and working inside Y[G] construct a tree A of height w with the nodes labeledso that an infinite branch of A corresponds to a pair (b, a), where b is a branchthrough k-1(S) and a is an order preserving map which witnesses that the limit

M;-l(S) of k-1(S) is well-founded at least up to k-1('Y). This means that either

a maps M;-l(S) into k-1b) or else a maps an initial segment of M;-l(S) ontok-1b)·

Now in V there is a branch of k-1(S) which is well founded up to k-1('Y) ,and hence in V there is an infinite branch through A. It follows that there is aninfinite branch through A in Y[G]. Thus k-1 (S) has a branch in Y[G] which iswell-founded up to k-1b). By the Martin-Steel theorem, Lemma 2, there can beat most one such branch and it follows from the homogeneity of the Levy collapsethat this branch is in Y . 0

Claim D. The model M is not in L[E].

Proof. Let b be the main branch of the tree I, and let C be the set of indiscerniblesgenerated by the embeddings in b, that is, the members of C are the ordinalse= crit(ir, 0) where v' E b.,

We claim that C C 8+ 1. Suppose to the contrary that there is v E C suchthat 8 < u < 8$. Then v is a regular cardinal in Mo, but is not a cardinal inW. As in the proof of Claim B, we can compare Mo with the least mouse in Wwhich has a definable map from 8onto u, Since both sides of the comparison havedropped to mice, the final models are equal, but this implies that v is collapsed inMo, contradicting the fact that every member of C is inaccessible in Mo.

Since C C 8+1, the set hM e "(P) is cofinal in 8. Either the model M is equal toult(Mo, rr, <5$) or else M = ult(M, n , <5$) where M is a mouse in Mo with projectumsmaller than 8; in either case there is p < 8 such that hM "p is unbounded. Since8 is regular in L[E] it follows that M ¢ L[E]. 0

Part two of the proof of the main theorem. We begin the second part of the proofby proceeding just as with the classical proof of the covering lemma, comparingthe models L[E] and M . Let I and U be the comparison trees on M and L[E], respectively. Notice that neither tree uses any extenders with critical point a smallerthan 8, since such an extender would give an iterable L[E]-ultrafilter on a < 8.

Claim E. If a is a cardinal such that 8 < a < K, then len(T) > a and i~a "a C a.Furthermore if K, is a cardinal or K, = Ord then leneT) = K, if K, < Ord and

L[Eu] has a Woodin cardinal, and len(T) = Ord otherwise.

Proof. To prove the first paragraph, let a be a cardinal with 8 < a < K,. SinceIMI =8 < a, the tree Ufa on L[E] can only involveextenders from Efa . It followsthat L[Eu ra] cannot be iterated to a Woodin cardinal and hence U always has awell-foundedmodel at least through Uf(a+1). Similarly, I must have wellfounded

192 W. J. MITCHELL!

models at least through a+1 since otherwise L[£nd would have a Woodin cardinalfor some ~ $ a, which is impossible since £n{ = £u t{. Thus len(T) ~ a + 1. SinceIMI < a = [o], the conclusion that i~a"a C a follows from the standard proof oftermination of comparisons.

If I\, is a cardinal or I\, = Ord then the proof of the comparison lemma againimplies that len(U) ~ 1\,. If len(T) > I\, then len(T) = Ord, since any extendersin £r which are longer than I\, cannot match any extenders in any model of U.If len(T) = I\, then L[£r] = L[£u] satisfies that there is a Woodin cardinal, sinceotherwise U and T would have final models, implying that M E L[£]. 0

Definition 8. Let a be a cardinal such that ~ < a < len(£), and let ba = [0, a]rand Ca = [0, a]u be the well founded branches through Tfa and Ufa respectively.We define Ca to be the set of ordinals ~ E ba n Ca such that

1. There are no drops in the branch [~,a]r, and i[a(~) = a .2. i~{(~) = ~ and crit(ir,a) ~ ~

The set of ~ E ba satisfying clause (1) is closed and unbounded, as is the set of~ E Ca satisfying clause (2). Hence the set Ca is closed, and if cf'(o) > w then Cais unbounded in a .

Clause (5) of the main theorem follows immediately: the members of Ca areindiscernibles in M! (apart for differing degrees of measurability, as described inthe statement of clause (5) of the main theorem) . Since P(a) n Mlj c M! itfollows that Ca is a set of indiscernibles for Mlj and hence for L[£].

Clause (4) is also immediate : if v E Ca and v > inf(Ca) then bv = ba nv andCv = Ca n v and it follows that O; = Ca n u,

Now let e= inf(Ca) and let v be in Co; . If we set C = (e+)M[ then i[,v"C iscofinal in (v+)M;; = (v+)M:;' . Since range(ir,v) is cofinal in (v+)M:;' it follows that

cf(v+L[e1) = cf(C) for all v E Ca , proving clause (2) of the main theorem.Then next two claims, which are clauses 1 and 3 of the main theorem, complete

the proof.

Claim F. If v E Ca , then v is weakly compact in L[£], and in fact

L[£] 1= I\, ~ (stationary)~.

Proof. If v E Ca then let E'J be the extender used at the vth stage in T, and letU; = (E'[)v be the associated ultrafilter. By amenability, U; is a MJ-ultrafilter.If we set U = {x E P(v) n L[£] : i~v(x) E Uv } then U is an L[£]-ultrafilter on v,and standard methods show that this implies that v is weakly compact in L[£].Furthermore, if f: [1\,]2 ~ 2 then the homogeneous set which is constructed is inU, and hence is stationary in L[£]. 0

Claim G. Suppose that v E Ca and cf(v+L[e1) > w. Then v is Ramsey in L[c],and in fact v ~ (stationary)<w in L[£].

Proof. Let U be as in the last claim. We will show that Un Ew iU" n X) is in L[c]whenever X E L[c] and IXIL[e1 = u. It follows (see [9] or [5]) that v is Ramsey

A HOLLOW SHELL 193

in L[f] , and as in the last claim the homogeneous set which is constructed is amember of U and hence is stationary.

We may evidently assume that X C P (Un Ew vn ) , so that X E L-yo [f] for some'Yo < (v+)L[t"l. Now for n ~ 0 let 'Yn+l be the least ordinal such that Un L-Yn[f] E

L-Yn+1 [fl· Then 'Yw = sUPn<w'Yn < (v+)L[t"] since cf(v+L[t"l) > w, and henceun L-y.., [f] E L[f]. This completes the proof since Un Ew (un n X) is constructiblefrom Un L-y..,[f]. 0

This concludes the proof of Theorem 5.

3. PROOF OF PROPOSITION 7

o

We wish to prove, under the hypotheses of the main theorem, that if 8 < Q < (3 < K,

and there are no measurable cardinals u of L[f] such that Q < u < (3 then all butfinitely many cardinals v in the interval Q < V < (3 either are Ramsey in L[f] orelse satisfy cf(v+L[t"l) = W .

Since there are no measurable cardinals in L[f] between Q and (3, the tree Irvin this interval consists of a linear iteration by the least measure above Q in themodels of I, and U is trivial in this interval. The critical points of the embeddingsin this interval form a closed set 0 which is unbounded in any cardinal v of V inthe interval. The iteration may drop (either to a mouse or in degree) finitely manytimes in the interval ; this gives us a finite sequence Q = Qo < Q1 < .. . < Qk = (3such that 0 0 ; + 1 = 0 n (Qi+1 \ Q i) whenever 0 :$ i < k. Thus any cardinal v in

(3 \ (Q U {Q1," . , Qk-1 }) is in OOi+1 for some ordinal Qi+l, and if cf(v+L[t"l) > wthen it follows by Theorem 5(3) that v is Ramsey in L[f] . 0

The next proposition shows that Proposition 7 cannot be strengthened byeliminating the finite set of exceptions. It shows that if a is any set of cardinalswhich can be preserved by stationary tower forcing then the cardinals in a can bepreserved by an extension which also gives their successors in the ground modeluncountable cofinality.

Proposition 9. Suppose that M = L[f] is an iierable model with a Woodin cardinal K" that 8 < 80 < .. . < 8n - 1 < K, are cardinals of M, and that there is acondition p in the stationary tower forcing over M which forces that

8=crit(k) and '<1m < n 8m is a cardinal

where k is a name in the forcing language for the generic embedding. Then thereis a condition p' which forces (*), and which also forces that cf(ijm) = W1 form =0 .. .n - 1, where TJm = (8;l;JM .

Proof. The hypothesis implies that M[G] F 18i l =8i ~ 8+(i+1) for each i < n , and

by basic facts about stationary tower forcing it follows that 8i has the followingJonsson property in L[f]:

If A is any structure in a countable language such that the universe A ofA contains 8i then there is an elementary substructure A' -< A such thatorder type(A' n 8i ) is a cardinal greater than or equal to 8+(i+1).

194 W. J . MITCHELU

By results in [12] it follows that 8i is 8+(i+l)-Erdos in M: that is, for everystructure A as above, and for every closed and unbounded C C 8i , there is anormal set of indiscernibles Dee for A of order type 8+(i+l) .

We will work in M for the rest of this proof; in particular all successors arecomputed in M. Let S be the set of y E P(8~_I) such that for each i < n we have

8 c y and ordertype(y n 8i ) =8+(i+l) and cf(y n 8t) = WI.

We will show that S is stationary (in the sense of stationary tower forcing) andhence (S,8~_I) is a condition. Then any generic set G' with (S,8~_l) E G' willsatisfy the requirements of the proposition.

We need to show that if A is any structure in a countable language with universeA = 8~_1' then A has an elementary substructure B with universe B E S. Wemay assume that the structure A has Skolem functions, so that it makes sense totalk about the Skolem hull 11:4 (X) of a subset X of A.

The proof of Proposition 9 is essentially the same as the proof that any 8-Erdos

cardinal is 8-Jonsson. We will define sets D, C 8i and b, C 8i+L[t:] for 0 :::; i < nsuch that if B = 1{A(8U Ui(Di Ubi)) then

1. order type(Di) =8+(i+l),2. o, is cofinal in B n s;3. order type(bi) =Wl ,4. bi is cofinal in B n 8t.

This will imply that B is the required elementary substructure of A.The sets D; and b, will be defined by a recursion on the integers i < n, taken

in descending order. Suppose that Dj and bj have been defined for i < j < nand set Ai+l = (A, Di+l' bi+l, ... ,Dn - l , bn - l). Since Ai+l has only finitely manypredicates in addition to those of A, it has a countable language and hence theset X of ordinals 13 such that 8i < 13 < 8t and 8t n 1{A; (13) = 13 is closed andunbounded in 8t . Choose f3i E X with cf(f3i) =WI, and let b, be any cofinal subsetof f3i of order type WI.

Now consider the structure Ai = (Ai+l, bi ) , and let D C 8i be the set of ordinalsv such that 8i n1{A; (v) = u. The structure Ai still has a countable language, andit follows that there is a normal set D, C D of indiscernibles for Ai such thatorder typefDr) =8+(i+I).

This completes the definition of the sets b, and Di. Clauses 1 and 3 are truefrom the definition, so we only need to verify clauses 2 and 4, which assert that biand D, are cofinal in B n8t and 8i respectively. To see that b, is cofinal in B n8t ,let A * = 1{Ai+l (13) and recall that 13 = sup b, was chosen so that 13 = 8t n A* .Since Wl < 8 < 13 and order typetbj) = Wl, we have bj C A* for i < j < n . Also,D j n A* is infinite, and since D j is a set of normal indiscernibles it follows thatB n 6t c A * . Since b, is cofinal in A * n 8t it follows that b, is cofinal in B n 8t.

A similar argument shows that D, is cofinal in Bn8i • This completes the proofof Proposition 9. 0

A HOLLOW SHELL

4. REMARKS AND QUESTIONS

195

As was pointed out in the introduction, the results of this paper are deficient inat least three areas : they do not really deal with Woodin cardinals and hence arelimited to the minimal model for a Woodin cardinal; they do not deal usefully withoverlapping extenders and hence only deal with failure of the weak coveringlemmaoccurring below the least potentially measurable cardinal; and their conclusionseems weak, even when the hypotheses are true. We will look at these three pointsin reverse order .

Can the conclusions be strengthened? It seems reasonable to guess that the hypothesis of the main theorem implies the existence of a set G in V which is L[£]generic for some variant of the stationary tower forcing. If this is true, then anycardinal in V must be sufficientlystrong in L[£] that it could have been preservedby a stationary tower extension:

Conjecture 1. Let the model L[£] and the ordinals 8 and", be as in the hypothesis to Theorem 5. Then 11+ is (1I+)L[t'LErdos in L[£] for all cardinals II with 8 < IIand 11+ ~ n,

Notice that by [12] this is equivalent to the assertion that 11+ is (1I+)L[t'LJ6nssonin L[£], which is exactly the condition required to use stationary tower forcing tomake the hypothesis of the conjecture true .

It may be that Conjecture 1 could be refuted by refuting the following conjecture:

Conjecture 2. Let L[£] be a model with a Woodin cardinal 8 but no mice whichiterate to a Woodin cardinal , and let G be generic overL[£] for the stationary towerforcing. If M be the least mouse of L[iG(£)] which is not in L[£] then G E L[£, M] .

A more plausible version of this conjecture might be that G nVeE L[£, M] foreach ~ < ~. Notice that vy·Ml is closed under sharps, since L[£, G] is a generic

extension of L[£,M] and VoL[t' .G) is closed under sharps.

Proposition 7 asserts that to obtain a sequence C of infinitely many cardinals IIsuch that cf«II+)L[t'I) > w we need to start with a sequence of Ramsey cardinals.On the other hand , if U is measurable then we can do the stationary tower forcingin such a way that there is such Prikry sequence C for U which has this property.The next conjecture asks whether the measurable cardinal is necessary:

Conjecture 3. Suppose that L[£] is as in Conjecture 1 and that C is an infinitesequence of singular cardinals such that (1I+)L[t'] < 11+ and cf(II+L[t')) > w foreach II E C. Then the first measurable cardinal of L[£] is not larger than sup(C) .

What happens above a measurable cardinal? The most straightforward generalization of the main theorem would be given by the following conjecture:

Conjecture 4. The main theorem is still true if the hypothesis that there are noiterable L[£] ultrafilters on any TJ < 8 is weakened to the hypothesis that there areno L[&] ultrafilters on any 1] < cf(~Ell)

196 W. J. MITCHELL!

The following example shows that the hypothesis that there are no iterableL[£] ultrafilters on any TJ < 8 cannot be simply eliminated: Suppose that V =L[£'] satisfies the hypothesis of Theorem 5, let U being the measure on the firstmeasurable cardinal p of V, and set i: V ~ L[£] =ultp+w (V,U) . Then 8 =p+w issingular in V, and cf((8+)L[E]) = p+ , but (TJ+)L[E] = TJ+ for all cardinals TJ > p+wofV.

The following conjectures are not ruled out by the example above:

Conjecture 5. If there is an iterable model with a Woodin cardinal, then there isan iterable inner model L[£] with a Woodin cardinal which satisfies the conclusionto Theorem 5.

Conjecture 6. Suppose L[£] is an iterable model and no mouse in L[£] can beiterated to a Woodin cardinal . If V is a set generic extension of L[£] then L[£]satisfies the conclusion to Theorem 5.

The following model may be of interest, though the requirement on the modelM is not satisfied by L[£], and is not even known to be consistent:

Suppose that M satisfies that 8 is a Woodin cardinal, witnessed by a set E ofextenders, and that M satisfies that £ is 8 + 1 iterable in M in the sense thatthere is an iteration strategy in M for all iteration trees in M which have lengthat most 8 + 1 and which use only extenders from £. Working in M, let U E E bethe smallest measure in £ and let Mf/ = ultf/(M, U), where TJ = I\:+w. Now let 7be an iteration tree of length 8 + 1, with last model N, such that U is N -genericfor Woodin's "all sets generic" forcing. If A = (l\:+w)N(U] then A ~ TJ, so (A+)Nis collapsed in N[U] . On the other hand there is a closed and unbounded subsetC C 8 in M of ordinals which are fixed by the embedding

i~ i TM --'--+ M; = ultf/(M, U) ---+ N.

If a E C then (a+)N = (a+)M = (a+)N[U] since N[U] C M . Thus it is nottrue that every successor in N of a singular cardinal of N[U] between A and I\: iscollapsed; and it is not even known whether any singular cardinal of N other thanA has its successor collapsed in N[U].

It is not even clear what happens in a nonstationary tower extension:

Conjecture 7. If 8 is a Woodin cardinal in M and G is generic for the stationarytower forcing up to 8 with crit(iG) < I\: < 8 then 1\:+ is not a cardinal in M[G].

The referee pointed out one piece of evidence in favor of this conjecture: Burkehas proved [1] that if 1\:+ remains a cardinal in a stationary tower extension M[G]for any I\: < 8 such that I\: is larger than the critical point of the generic embedding,then Jensen's property 0" fails in M. On the other hand Schimmerling [16] showedthat the weaker property O~w is true for all cardinals I\: in L[£].Thus the conjecturewould be proved if Burke's result were improved to O~w or if Schimmerling's resultwere improved to 0".

What happens past a Woodin? The first problem in extending the results of thispaper to larger cardinals is the construction of the necessary inner models, so the

A HOLLOW SHELL 197

following conjecture appropriately fits here though it really concerns the situationbelow a Woodin cardinal:

Conjecture 8. If there is no iterable inner model with a Woodin cardinal thenthere is an inner model K, which is preserved in set generic extensions, such thatthe weak covering lemma holds over K.

There is a good candidate for the desired inner model: the Steel core model,constructed from below. At the present time, however, the only known way of verifying that this model is iterable is by using Steel's "certified" core model KC, whichrequires something approaching an extra measurable cardinal for its construction.

A partial answer given by a construction of Woodin (see [17]) is not entirelysatisfactory, particularly below a Woodin cardinal. Woodin's core model has theform K v, for some ordinal a. Thus it contains all of Va and gives no informationabout the universe below a, which is likely to be exactly the region that we areinterested in.

A final, and the most important question, is what cardinal strength is neededin order to have more control over which cardinals are collapsed. Suppose that", is",+-supercompact, and let P be Prikry forcing using a ",+-supercompact measureon n, Then forcing with P will change the cofinality of both", and ",+ to w,and will collapse ",+ in the process, but will not otherwise change cardinalities orcofinalities. Is the ",+-supercompact cardinal needed?

Conjecture 9. Suppose that M is a model of set theory with a singular cardinal8, and M[G] is a generic extension of M such that 18+(M) I = 8 but all othercardinals of M are preserved in M[G]. Then there is an inner model with a 8+supercompact cardinal.

References

1. Douglas Burke, Generic embeddings and the failure of box, Proceedings of theAMS 123(9) (1995) 2867-2871.

2. Keith Devlin and Ronald B. Jensen, Marginalia to a theorem of Silver, in:[SILC Logic Conference (Kiel 1974), Springer-Verlag, Berlin and New York(1974) 115-142 .

3. Anthony Dodd, The Core Model, London Mathematical Society Lecture NoteSeries 61, Cambridge University Press, Cambridge, England, 1982.

4. Matt Foreman, Menachim Magidor, and Saharon Shelah, Martin's maximum,saturated ideals, and non-regular ultrafilters, I, Annals of Mathematics (2ndseries), 127 (1988) 1-47 .

5. James Henle and Eugene Kleinberg, A flipping characterization of Ramseycardinals, Zeitschrift fur Math . Logik und Grundlagen der Math, 24 (1) (1978)31-36.

6. Ronald B. Jensen, The fine structure of the constructible hierarchy, Annals ofMathematical Logic 4 (1972) 229-308.

7. Kenneth Kunen, Some applications of iterated ultrapowers in set theory, Annals of Mathematical Logic 1 (1970) 179-227.

8. Donald A. Martin and John R. Steel, Iteration trees, Journal of the AmericanMathematical Society 7 (1) (1994) 1-73.

9. William J . Mitchell , Ramsey cardinals and constructibility, Journal of SymbolicLogic 44 (2) (1979) 260-266.

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23.

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William J. Mitchell, The core model for sequences of measures, MathematicsProceedings of the Cambridge Philosophical Society 95 (1984) 41-58.William J. Mitchell, The core model of sequencesof measures, II, 1984 preprint.William Mitchell, Jonsson cardinals, Erdos cardinals, and the core model, 1994preliminary version, available on Logic Eprints.William J. Mitchell and John R. Steel, Fine Structure and Iteration TI-ees,ASL Lecture Notes in Logic 3 Springer Verlag, 1994.William J . Mitchell, E. Schimmerling, and John R. Steel, The covering lemmaup to one Woodin cardinal , Annals of Pure and Applied Logic 84 (2) (1997)219-255.Karl Prikry, Changing measurable into accessible cardinals, DissertationesMathematicae (Rozprawy Mathematycne) 68 (1971) 359-378.Ernest Schimmerling, Combinatorial principles in the core model for oneWoodin cardinal , Annals of Pure Applied Logic 14 (1995) 153-201.Ernest Schimmerling and W. Hugh Woodin, The Jensen covering property,The Journal of Symbolic Logic, to appear .Ernest Schimmerling and John R. Steel, Fine structure for tame inner models,Journal oj Symbolic Logic 61 (1996) 621-639.John R. Steel, Core models with more Woodin cardinals, 1993 preprint.John R. Steel, Inner models with many Woodin cardinals, Annals oj Pure andApplied Logic 65 (2) (1993) 185-209.John R. Steel, The Core Model Iterability Problem, Lecture Notes in Logic 8,Springer Verlag, Berlin, 1996.W. Hugh Woodin, Supercompact cardinals, sets of reals, and weakly homogeneous trees, Proc. Nat . Acad. Sci. U.S.A . 85 (18) (1988) 6587-6591.W. Hugh Woodin, The axiom of determinacy, forcing axioms and the nonstationary ideal, Book in preparation.

PARTITION PROPERTIES FOR REALS

CARLOS H. MONTENEGRO1

Universidad de Los Andes, BogotaMatematicas, Universidad de Los Andes, A .A . 4976,Bogota, Colombia

Abstract. We look at a partition relation viewed as a generalization of HalesJewett's theorem. By making modifications to this partition relation , we describesome results and problems, and end in the full polarized partition relation ofDi Prisco and Henle [2].

1. Introduction

The study of partition relations deals with mappings X : A ~ C of a given structureA on a set A, into a set C. In this context , the mappings X are colorings and C isa set of colors (usually a finite set) . Examples of structures on A are: a cartesianpower of A, the set of its n-tuples, simple graphs with vertex set A, a well orderof A, etc .. Given such coloring X : A ~ C , we are looking for special (large)subsets H ~ A, on which the mapping X : AI1-l ~ C is as simple as possible. Inthis context, it is natural to ask for a subset H ~ A with IHI = IAI, where f isconstant on Alll. In such cases, the set H is called a homogeneous set for X.

When presented with a partition relation , one is given three parameters:

1. the structure type,2. the sort of homogeneous set desired, and3. the set of colors C.

The partition relation is an assertion using the above parameters: for any structure A of the given type, for any coloring X : A ~ C, there is a homogeneousset H ~ A for X of the desired sort . For example, to get the partition relationw ~ (w)W, we take for (1) [w]W , the set of all infinite subsets of w, for (2) infinitesets and for (3) a two element set . So we get the statement: for every X : [w]W ~ 2there is an infinite set H, such that X is constant on [H]w . Sometimes a fourthparameter is given specifying the type of coloring X allowed. For example, take for

IThe author gives special thanks Carlos Di Prisco and the mathematics departmentof IVIC for the hospitality during his visit from February to June 1995.

199

C.A. Di Prisco et al. (ed.), Set Theory, 199·205.© 1998 Kluwer Academic Publishers.

200 CARLOSH.MONTENEGRO

(1) infinite subsets of w, for (2) an infinite subset, for (3) a finite set and ask thatthe colorings be Borel. This gives Galvin-Prikry's [4] result.

2. Products

For the structure, we are interested in using nonempty products IIiEIXi whereXi ~ wand I ~ w. Given a nonempty product P = IIiE1Xi, its type is thefunction I : w -t w given by I(i) = IXil- 1. The support of P is the support ofits type I, i.e, the set 8 = {i E I : IXil > I} . Given a family 11. of functions , wesay that a product IIiEIHi is of type 11. if its type is one of the functions in 11.. Wework with the following parameters:

1. The structures A are products IIiEIXi where Xi ~ wand I ~ w.2. The homogeneous sets are specified by giving a family 11. ~ W W of possible

types.3. A finite set of colors (usually 2).

We show how this partition properties can be viewed as a natural variationto Hales-Jewett's theorem, and make modifications to end in the full polarizedpartition relation of Di Prisco and Henle [2].

3. Hales-Jewett Theorem

Our starting point is the Hales-Jewett theorem, that we rephrase in this context.Hales-Jewett's theorem is a partition property where,

1. the structure is a finite power TN of a finite set T with t elements.2. the homogeneous sets H (the combinatorial lines) are contained in a product

P = IIiENHi, where for each i E N, the size IHil of each set is either 1 or tand the set 8 = {i : IHil = t} of P is not empty. A function f E P is in thehomogeneous set H if and only if liS is constant.

3. a finite set of colors with c elements.

The one dimensional Hales-Jewett's is an existence theorem stating that forany set T of t E w elements, and any finite C with c E w colors, there is an NEwsuch that the partition property with the above parameters holds (it actually statesmore, but this is what is relevant to us) . The m dimensional version allows for thesupport S = {i : IHd = t} in (2) above, to be partitioned in m sets 8o, ... 8m - 1

and any function I E IIiENHi that is constant in S; for each i E m is in thehomogenous set. The number N guaranteed by Hales-Jewett's theorem is denotedby N = HJ(t,m,c).

No other conditions (besides being nonempty) can be imposed on the supportset 8, i.e., we cannot impose cardinality conditions (other than lSI> 0) nor canwe specify an element i E S. To see about the cardinality condition, if we are givena k E w, we define the coloring X(J) = ((J(O) +.. .+I(N -1))(mod(k-1))(modc) .This coloring can not have a homogeneous set with support set 8 having lSI = kbecause, since I can be taken to be constant on S, the contribution to the sumof I on S is its constant value (modulo k - 1). So its color changes as this value

PARTITION PROPERTIES FOR REALS 201

changes (notice that to change this color, it suffices to increase or decrease thisvalue by one). It is easy to see that we cannot specify any elements i of S (justcolor f by f(i)(modc) ).

We are interested in having the homogeneous sets to be the whole productITiENHi (instead ofjust being contained in the product), and by the above remarks,we cannot ask that any Hi have two consecutive values, and in particular, Hicannot have size greater than (t + 1)/2. What we do to get around this is toremove the cardinality conditions on the sets Hi and simply ask that the supportY ={i : IHil > I} be non-empty (instead of asking that, for each i in the support,the set Hi is of maximal possible size).

With this modification to Hales-Jewett's theorem, we have a partition property P(t, n,m, c) stating that: given natural numbers t, n, m, c and any coloringX : t" -t c there is a homogeneous set ITiENHi with Hi: IHil =2}1 =m.

For simplicity, take the case c =2. Clearly, if m =0 the partition always holds(just take any function), and it is false for m > n, so we assume that 0 < m 5 n.Also note that the partition is false for 2 = t and m > 1 (as this is the Hales-Jewettversion), and so we also assume 2 < t.

Varying only m, the strongest partition property is the case m = n where allthe sets Hi have size 2. As we show bellow P(t, n ,n, 2) fails for t < 2n and it istrue for R(n, 2n) < t where R(n,2n) is the least N such that N -t (2n)~ (i.e., theleast N such that any partition of the n-sets of N in two colors has a homogeneousset H of size 2n) .

Example 1. P(t, n , n , 2) fails for t < 2n . Consider the coloring X such that xU)is the parity of the size of the range of f . Let ITiEnHi be a homogeneous set . Sincet < 2n the Hi are not disjoint. Let y E Hi n Hj with i i j and take any f inITiEnHi with f(i) = f(j) = y . Since all the Hi have two elements, we can changef, to another functions in the homogenous set, by changing its value at any xwith f(x) = y. Note that changing anyone of these values gives a function in thehomogenous set with the same range as f (because y remains in the range of thenew function, and since the parity of the size of the range of this new functionmust not change, this new value must be in the range of 1). But this cannot bebecause when changing all these values, y will no longer be in the range and therange decreases by 1 changing the color of the function.

Example 2. If R(n, 2n) < t then P(t, n, n, 2). Suppose we are given a X : t" -t 2.Define a coloring X of the n-sets of t by coloring the n-set X with the colorof the increasing function f E t" with range X. By Ramsey's theorem, thereis a homogenous set H = {ho, ... , h2n-1 } . Any function in ITiEn{h2i, h2i+d isincreasing and its range is an n-set of H, so this product is a homogeneous set as

desired. This property is expressed by

tt

~n

2 l,l, .. .~n...,l

2

~n

t 22

The analog of Hales-Jewett statement with this partition relation is that forany 2 < t and 0 < m there is an n such that P(t, n, m, 2), which is not always true(for eXanIple, it fails for t = 3, m = 2).

4. Finite products of infinite sets

Now we take finite products of w. In this case the results are simple consequencesof Ramsey 's theorem. We consider two properties as examples, In the first, we askfor homogeneous sets that are products of sets {Hi : i E n} with IHn-d = wandIHi I = 2 (or any finite cardinality) for i = 0, . .. ,n - 2. To see that this propertyholds proceed as in Example 2. Suppose we are given a coloring X : wn ~ 2. Colorthe n-sets of w by giving such set x the color that X assigns to the increasingfunction with range z. Get a homogeneous infinite set by Ramsey and use its first2(n - 1) elements for the sets Hi for i = 1, ... ,n - 2 and the rest for the set Hn-l .

Example 3. Now consider the case where we request that the homogenous setsbe products with at least two of its factor infinite. This property is false as canbe verified by the coloring X : wn ~ 2 that assigns to a function f the values 0 ifand only if it has exactly two largest values and they are increasing. Clearly thiscoloring cannot have homogenous sets with two infinite factors because the orderof the two largest values of a function can be changed by changing values withinthese sets.

5. Infinite products

Given a product IliEwXi we will be interested in homogenous products that arerequired to have infinite support (if the homogenous sets can have finite support,then this will be the case of the previous section). Using the axiom of choice itis easy to construct a coloring X with no homogeneous sets by well ordering allpossible homogeneous sets and defining the coloring X inductively on this well order, making sure that each possible homogeneous set gets functions of two colors.There are two approaches to the study of these partition relations, we can limitthe type of colorings allowed or we can investigate the consistency of the partitionproperties with ZF. In the first direction Moran-Strauss [9] have shown that ifX : ITiEI X i ~ 2 has the property of Baire, and each Xi is finite, then the homogeneous sets will be of the form ITiEwYi where Yi = Xi for infinitely many i E w.They also showed that if each X i =w then the homogeneous sets will be a product

PARTITION PROPERTIES FOR REALS 203

lliEWYi with Yi =w for finite many i. Using the fact that there are models of ZFin which every set of reals has the property of Baire [10] we get the consistency ofthese partition properties with ZF.

The situation is quite different if we require that the homogeneous lliEwHisets have full support w (I.e. (Vi E w)(IHil > 1). For example, if Xi = T E wfor all i E w, then the partition property fails (color f by the parity of the rangeof fl(2T) and argue as in Example 1). Martinez [8] gave an example of a Borelcoloring X : WW -t 2 such that no function of the form f (k) = A + k for A E w,dominates all h E lliEwHi. This shows that if there is a {XihEW such that anycoloring X: lliEwXi -t 2 has homogeneous sets ll iEwHi having full support, thenIXil cannot be dominated by a linear function f(i) = A + i . However, Llopis andTodorcevic [6] have shown that if we restrict to Borel colorings, there is a productof finite sets lliEwXi for which the partition property holds. Their result states,more generally, that given any function f E WW there is a product lliEwXi of finitesets for which the partition property hold with homogeneous sets of type f .

In the case where Xi = w the partition property requiring the homogeneoussets to have full support is an easy consequence of w -t (w)w. Given a coloringX : WW -t 2 define the coloring on the infinite subsets Y ~ w by assigning to Ythe color that X assigns to the increasing function with range Y, and argue asin Example 2. Note that the homogeneous set can be obtained to have any typef E ur" , Much of the recent work in this area was motivated the asking [2] ifw -t (w)W is a consequence of this partition relation. There is strong evidence thatthese is not the case [6]. A strategy to show this, first suggested by Di Prisco, isto prove that this partition relation is consistent with the existence of ultrafiltersin w [1] . This would settle the problem since one can construct counterexamplesto w -t (w)W using an ultrafilter [7] .

Now consider the case where the type of the homogeneous set is a functionf : w -t (w + 1) with infinite support (i.e, we might require the homogeneous setsto have infinite sets in its product). Henle [5] has shown that if w -t (w)W holds,then given any X : WW -t 2 there is a {HihEW with IHil > 1 for all i E w andone of the His infinite (one can even choose a k E w and find the set {HihEWwith Hk infinite). If we choose two values a < b, it is easy to construct a coloringX : W W -t 2 that cannot have any homogeneous products with Ho. and Hb bothinfinite (proceed as in Example 3). It is not known to be consistent that anycoloring X :WW-t 2 has a homogeneousset lliEwHi with more than two of the Hiinfinite [2] . However, Di Prisco and Henle [2] have given an example of a coloringthat cannot have homogeneous sets lliEwHi with infinitely many Hi infinite. It isinteresting that almost all known counterexamples to partition relations of this sorthave the property that one can change the color of a function f by making finitelymany modifications to f. This case is an exception, there is a Fin-invariant coloringX : WW -t 2 such that if lliEwHi is homogeneous for X, then the set {i : IHil =w}is a finite set [8]. We know very little about these type of partitions when werestrict to Fin-invariant colorings, for example, is the existence of an ultrafilter inw enough to find a Fin-invariant counterexample to w -t (w)W?

6. Stronger partition properties; polarized partitions

Henle's result [5] can be viewed as the case a = 1 in the following partition relationPP(a),

1. The structure is [w]Q x "t», where a ~ w and [w]Q are the subsets of w of sizea.

2. The homogeneous sets are of the form H x (TIiEwHi) where H is an infiniteset and IHil > 1 for all i E w.

3. Two colors

Recently, Llopis and Todorcevic [6] have announced that the strongest partition(i.e., PP(w)) holds for Borel colorings. The consistency of this partition relationwill be enough to follow Di Prisco's strategy and show that the existence of anultrafilter in w is consistent with partitions of ww that require homogeneous set offull support [1].

It turns out that the parametrized partition relation PP(w) is equivalent tothe following partition C,

1. Functions f E Co ={f E Ww : {n : f(n) =O} infinite} .2. Homogeneous sets are products TIiEwHi with IHil > 1 having functions in Co·3: Two colors

Theorem 6.1. PP(w) if and only if O,

Proof. (=» . Suppose PP(w) and let x: Co ~ 2 be given. For each (X,I) in[w]W x Ww define a function fx E Co by setting fx(x) =0 if x E X, and fx(x) =f(x) + 1 otherwise. Now define x: [w]W X Ww~ 2 by setting X(X, J) = xUx).

Let [A]W x (TIiEwFi) be an e-homogeneous set for X (i.e., for all (X, I) E[A]W x (TIiEwFi), X(X, I) = f) . Suppose that Fi = {ai, bi} with ai < bs, anddefine the sets Hi for i E w by setting Hi = {O, bi + I} if i E A, and lettingHi = {ai + 1, b, + I} otherwise.

If hE TIiEwHi then X ={i : h(i) =O} ~ A (since 0 E Hi if and only if i E A).Let h' be defined by h' (x) = b; if x E X and h' (x) = h(x) - 1 E {ai, bi} if x rt X.Notice that h' E TIiEwFi and h~ = h. So X(h) =X(h~) = x(X, h') = e and thusTIiEwHi is an e-homogeneous set for X as desired.

C~) Let X : [w]W x Ww ~ 2 be given. For each a E Co let ai(n) = a(2n + i)and Xf = {n : ai(n) = O} for i =0,1. One of the Xf must be infinite. Define acoloring X: Co ~ 2 by,

{

0x(a) = 1

x(Xf,al-i)

if Ixgl = IXf l =wand nxg < nXfif Ixgl = IXfl = wand nxg > nXfif IXfl =w, IXf-il < w

Let TIiEwFi be an e-homogeneous set for X, and suppose F; = {ai, bi} with ai < bi·Let a(n) = an, Note that xg, Xf cannot both be infinite (as we can modify ato a f3 in TIiEwFi such that n xg < n Xf ¢} n xg > n xf changing f). Supposexg is infinite, and define H = {n/2 : n E xg A 0 E Fn}, and Hn = F2n+l forn E Xf. We need to verify that [H]W x TIiEwHi is an e-homogeneous set for X.

PARTITION PROPERTIES FOR REALS

Let (Y,h) E [H]Wx TIiEwHi. Define "I E Co by,

205

obk

h(k)

n = 2k,k E Yn = 2k ,k st Yn = 2k + 1

Note that XJ =Y and "11 = h and so X(Y,h) = X(XJ,"I1) = X("() = e, the lastequality since "I E TIiEwFi. 0

There are many interesting (and difficult) open questions in the subject [3] .

References

1. C. Di Prisco, Partition Properties for Perfect Sets, Notas de L6gica Matematica,INMABB-CONICET, Bahia Blanca, Argentina, 38 (1993) 119-127.

2. C. Di Prisco and J. Henle, Partitions of Products, Journal of Symbolic Logic 58(1993) 860-871.

3. C. Di Prisco and J. Henle, Partitions of the Reals and Choice, (preprint) .4. F. Galvin and K.Prikry, Borel Sets and Ramsey's Theorem, Journal of Symbolic

Logic 38 (1973) 193-148.5. J. Henle, One the Consistency of One Fixed Omega, Journal of Symbolic Logic 60

(1995) 172-1776. J. Llopis and S. Todorcevic, A Polarized Partition Relation for Borel Sets (preprint).7. A.R.D. Mathias , Happy Families, Annals of Mathematical Logic, 12 (1977) 59-111.8. M. Martinez and C. Montenegro, Product Spaces, (preprint).9. G. Moran and D. Strauss, Countable Partitions of Products, Mathematika 27 (1980)

213-224.10. S. Shelah Can You Take Solovay's InaccessibleAway? Israel Journal of Mathematics

48 (1984) 1-47.

COMBINATORIAL SET THEORY AND INNER MODELS

E. SCHIMMERLINGlDepartment of MathematicsUniversity of California, IrvineIrvine, California, 92697USA

Abstract. Core model theory has lead to a new hierarchy of square principles. Inthe other direction, various combinatorial principles have been used to constructcore models with Woodin cardinals.

R.B. Jensen's principles OK and O~ have played key roles in both Inner ModelTheory and Combinatorial Set Theory. In this survey, we intend to introduce thepractitioner of the latter to a new hierarchy of intermediate principles that arosein the study of inner models. One of the most important consequences of Jensen'sfine structure theory for L is that for any singular cardinal K" if 0# does not exist ,then OK holds in V. This fact has been used to obtain many relative consistencyresults at the level of 0#. Here, we shall describe recent progress in extending thismethod to the level of Woodin cardinals.Definition 1. ([13,5.1]) Suppose that K, and A are cardinals. By O~A we meanthat there is a sequence (Fv I v < K,+) such that whenever K, < v < K,+ and v is alimit ordinal:

(1) 1 s card(Fv ) < A(2) for every C E Fv :

(a) C is a club subset of v

(b) o.t.(C) s K,

(c) J.L E lim(C) ==> C n J.L E Fp.

And, also, we shall write O~ for O~A+.

Note that OK and O~ are equivalent, as are O~ and O~ . It is easy to see thatOw and O~+ hold. Also that if K,<K = K" then O~ holds. Jensen proved in [8] that

IThis paper was written while the author was a National Science Foundation postdoctoral fellow at the Massachusetts Institute of Technology.

The author thanks Carlos DiPrisco and Jean Larson for having organized a superbconference in Curacao.

207

208 E. SCHIMMERLING

E= (Eo: I a E OR) ,

,ON! + O~! is consistent relative to a Mahlo cardinal. Many other results thatdistinguish O~>' from O~ are known, but we shall not attempt to catalogue themhere. (See [2] and [3].)

Jensen [6] showed that OK (and even stronger principles) hold in L. Let usdescribe the models to which we would like to extend Jensen's theorem. We shallbe vague about the details , since they are numerous, and we shall suppress someentirely. By

we shall always mean a "good" extender sequence, in the sense of W.J. Mitchelland J.R. Steel, as introduced in [12]. The universe constructed from E, denotedL[E], has what are called "levels", or structures of the form

for a ~ OR. In what is called the "active case" , there is some JL < a such that Eo:is a (JL , a)-extender over J! .So

Eo: = (EO:)a Ia E [a]<W)

is a directed system of ultrafilters, with each (Eo:)a an ultrafilter on J! np(lJL]lal).This ordinal JL will be denoted by crit(Eo:). In the "passive case", Eo: = 0. Astructure M is called a "premouse" if M is of the form :r! (for some a ~ OR)and every level of M is sound (which means that M satisfies a strong form ofGCH). A premouse M is called a "mouse" if M is iterable in a sense involvingiteration trees that we shall not make precise. Also, a premouse M is said to be"I-small" if for all f3 ~ a,

E{J '" 0 =} JII= "there are no Woodin cardinals".

A mouse that is not l-small corresponds, in a natural way, to indiscernibles foran inner model with a Woodin cardinal. We remind the reader that a Woodincardinal has consistency strength between that of a strong cardinal and that ofa superstrong cardinal. In [12, 11.3], Mitchell and Steel showed that if there is aWoodin cardinal, then there is a mouse with a Woodin cardinal. Building on thefine structure for mice developed in [12] and [13], the author showed:

Theorem 2. ([14]) If M is a l-small mouse, then M 1= 'tIK. O~w .

We remark that the methods of [8] can be used to see that O~;+'tin < W ,ON!is consistent relative to a Mahlo cardinal.

In order to use Theorem 2 to obtain relative consistency results, we shall needsome covering theorems. Of course, these build on the well-known results of Jensen,A.J . Dodd, and Mitchell; see [4], [9], and [7] . Steel developed a core model theoryat the level of one Woodin cardinal in [19], beginning with the following "cheapo"covering theorem.

Theorem 3. ([19, 1.4]) Suppose that there is no inner model with a Woodincardinal and that K. is a measurable cardinal. Then there a premouse KC of heightK. such that (a+)K

c =a+ for almost every a < K..

SET THEORY AND INNER MODELS 209

KC is Steel's "background-certified core model" . An immediate consequence ofTheorems 2 and 3 is:

Corollary 4. Suppose that there is no inner model with a Woodin cardinal andthat K is a measurable cardinal. Then O~w holds.

A well-known argument due to R.M. Solovay can be modified to yield thefollowing result, as was shown to the author by A. Kanamori and D. Burke; see[18].

Theorem 5. (Solovay) Suppose that K ::; A are cardinals such that K is A+·strongly compact. Let J.t = cJ()). Then O~J.l fails. Moreover, if J.t < K, then O~fails.

From Corollary 4 and Theorem 5 we conclude that if there is a cardinal K thatis K+-strongly compact, then there is an inner model with a Woodin cardinal. Astronger statement is true, as we shall explain soon .

Corollary 4 should also be viewed in conjunction with S. Todorcevic's result in[22], that the Proper Forcing Axiom (PFA) implies that OKfails for all uncountableK . Todorcevic's argument can be modified to show the following, as was shown tothe author by M. Magidor; see [13, 6.3].

Theorem 6. (Todorcevic) If PFA holds and K > N1 , then O~l fails .

More recently, Magidor proved that if PFA is consistent, then so is PFA + VKO~2 . 2 Recall that J. Baumgartner and S. Shelah showed that PFA is consistentrelative to a supercompact cardinal. From Corollary 4 and Theorem 6, we concludethat if there is a measurable cardinal and PFA holds , then there is an inner modelwith a Woodin cardinal. By [13, 6.5], the measurable cardinal is not needed forthis lower bound on the consistency strength of PFA; part of the argument usesthe following covering result of Mitchell and the author.

Theorem 7. ([10, 0.1]) Suppose that there is no inner model with a Woodincardinal and that K is a measurable cardinal. Suppose that 0: is an ordinal suchthat W2 ::; 0: < K . Then there is a premouse K such that cf(o:+)K) ~ card(o:). Inparticular, if 0: is a singular cardinal < K, then (o:+)K =0:+.

K is Steel's "true core model", which he introduced in [19]. Theorem 7 buildsdirectly on the main result in [11], which is the same except that card(o:) is assumedto be countably closed. Steel showed in [19, 8.14] that K computes successors ofmeasurable cardinals correctly (assuming a measurable above) . More recently, theauthor showed that K computes successors of weakly compact cardinals correctly(assuming a measurable above); see [16]. From this, Theorem 7, and Theorem 2,we conclude:

Corollary 8. Suppose that there is no inner model with a Woodin cardinal andthat K is a measurable cardinal. Then O~w holds for every cardinal 0: < K whichis either singular or weakly compact.

2Magidor also proved that Martin's Maximum (MM) implies that O~ fails for allcardinals It of countable cofinality.

210 E. SCHIMMERLING

Let us mention some recent improvements to Corollary 8. First, the authorobserved that the hypothesis that /'i, is a measurable cardinal can be weakened to

in Corollary 8; see [16]. Similar partition relations suffice in many other applications of the core model. Second, Steel and the author [16] showed that if O~w failsand a is either a weakly compact or a singular, strong limit cardinal , then there isan inner model with a Woodin cardinal (without assuming that there is a measurable cardinal above a). Building on the latter result, W.H. Woodin showed thatif a is a weakly compact cardinal and O~w fails, then L(IR) determinacy holds.Similarly, if a is a singular, strong limit cardinal and O~w fails, then ProjectiveDeterminacy (PD) holds.

An attractive corollary to the aforementioned work of Mitchell, Steel, Todorcevic, Woodin, and the author is:

Theorem 9. Suppose that PFA holds. Then PD holds. If, in addition, there is aninaccessible cardinal, then L(IR) determinacy holds.

Of special interest is Nw • By Corollary 8, if there is a measurable cardinaland O~ fails, then there is an inner model with a Woodin cardinal. In the otherdirection, it is known that if there is a cardinal /'i, which is /'i,+-supercompact, thenthere is a model in which ON... fails; see [17] (and, also, [1]). While the consistencystrength of ""0N... ,or even ...,ON... ,has been conjectured to be close to this upperbound, we still do not know the answers to some basic questions.

Problems 10.

ra) If there is a measurable cardinal and ON... fails, is there an inner modelwith a Woodin cardinal?

(b) If ON... fails, is there is an inner model with a Woodin cardinal?(c) IfNw is a strong limit cardinal and ON... fails, does L(IR) determinacy hold?(d) If there is a measurable cardinal and ON... fails, is there an inner modelwith two Woodin cardinals?

We turn now to generalizations to larger large cardinals, beyond those corresponding to L(IR) determinacy. As in [20, 0.2], we shall say that a premouse .7!is "meek" if for all (3 ~ a,

E{3 '" 0 ==} J::"it(Ep) 1= "the Woodin cardinals are bounded".

A mouse that is not meek corresponds to indiscernibles fora model with a properclass of Woodin cardinals . In [20] and [21] , Steel extends much of the theory of[12] and [19] to the level of meek premice. Corresponding extensions of the resultsin [13, 14, 11, 10] also hold.

Continuing on to even more powerful premice, following [20, 0.1], we say thata premouse .7! is "tame" if for all (3 ~ a

E{3 '" 0 ==} J! 1= V7 ~ crit(E{3) "7 is not a Woodin cardinal".

SET THEORY AND INNER MODELS 211

If there is a premouse that is not tame, then there is an inner model with astrong cardinal that is a limit of Woodin cardinals (and more). But the existenceof a mouse that is not tame does not imply that there is an inner model with ameasurable Woodin cardinal. In [20] and [15], enough of the core model theory isextended to prove the following result of Steel and the author.

Theorem 11. ([15]) Suppose that", is a measurable cardinal and that all premiceare tame. Then there is a premouse K C of height", such that:

(aJ K C 1= Va O~w

(b) (a+)Kc = a+ for almost every a < '"

But we do not know if the tame KC is sufficiently iterable to develop the theoryof the tame K . Thus, Theorem 11 falls short of a satisfactory core model theoryfor tame premice. Nevertheless, we do get an extension to Corollary 4, and somerelative consistency results at the level of tame premice.

Corollary 12. If « is a measurable cardinal and all premice are tame , then O~w

holds.

Corollary 13. If there is a cardinal", that is ",+-strongly compact, then there isa premouse that is not tame .

Corollary 14. If PFA holds and there is a measurable cardinal, then there is apremouse that is not tame .

References

1. S. Ben-David and M. Magidor, The weak O' is really weaker than the full0, Journal of Symbolic Logic 51 (1986) 1029-1033.

2. J. Cummings, M. Foreman, and M. Magidor, preprint.3. J . Cummings and E. Schimmerling, in preparation.4. K.I. Devlin and R.B. Jensen, Marginalia to a theorem of Silver, in: Logic

Conference, Kiel1974, Lecture Notes in Mathematics 499 , Springer-Verlag,Berlin, (1975) 115-142.

5. A.J . Dodd and R.B. Jensen, The covering lemma for K, Annals of Mathematical Logic 22 (1982) 1-30.

6. R.B. Jensen, The fine structure of the constructible hierarchy, Annals ofMathematical Logic 4 (1972) 229-308.

7. R.B. Jensen , Non overlapping extenders, circulated notes.8. R.B. Jensen, Some remarks on 0 below 0' , circulated notes.9. W.J. Mitchell, The core model for sequences of measures I, Mathematical

Proceedings of the Cambridge Philosophical Society 95 (1984) 229-260.10. W.J. Mitchell and E. Schimmerling, Weak covering without countable clo

sure, Mathematical Research Letters 2 (1995) 595-609.11. W.J . Mitchell, E. Schimmerling, and J .R. Steel, The covering lemma up to

a Woodin cardinal, Annals of Pure and Applied Logic 84 (1997) 219 - 255.12. W.J . Mitchell and J.R. Steel, Fine structure and iteration trees, Lecture

Notes in Logic 3, Springer-Verlag, Berlin, 1994.13. E. Schimmerling, Combinatorial principles in the core model for one Woodin

cardinal, Annals of Pure Applied Logic 74 (1995) 153-201.14. E. Schimmerling, O~w holds in LIE], J. Symbolic Logic, to appear.15. E. Schimmerling and J .R. Steel, Fine structure for tame inner models, Jour

nal of Symbolic Logic 61 (1996) 621-639.

212

16.17.

18.

19.

20.

21.22.

E. SCHIMMERLING

E. Schimmerling and J.R. Steel, The maximality of the core model, preprint.S. Shelah, On successors of singular cardinals, in: Logic Colloquium '78(M. Boffa et al., editors), North-Holland, Amsterdam and New York, (1979)357-380.R.M. Solovay, W.N. Reinhart, and A. Kanamori, Strong axioms of infinityand elementary embeddings, Annals of Mathematical Logic 13 (1978) 73116.J .R. Steel, The core model iterability problem, Lecture Notes in Logic 8,Springer-Verlag, Berlin, 1996.J .R. Steel, Inner models with more Woodin cardinals, Annals of Pure andApplied Logic 65 (1993) 185-209.J.R. Steel, Core models with more Woodin cardinals, preprint.S. Todorcevic, A note on the Proper Forcing Axiom, Contemporary Mathematics 95 (1984) 209-218.

DEFINABLE IDEALS AND GAPS IN THEIR QUOTIENTS

STEVO TODORCEVIClDepartment of MathematicsUniversity of TorontoToronto, Canada M5S 3G3

andMatematicki InstitutK neza Mihaila 3511000 Beograd, Yugoslavia

Abstract. We extend some known results about analytic gaps and ideals onN to all definable gaps and ideals assuming a covering property of definablesets of reals X which says that if X cannot be covered by countably manymembers of some family F of closed sets then there is a G5 set with thesame property which is included in X .

The purpose of this note is to extend some known results about analytic gaps and ideals to all definable gaps and ideals either assuming somestandard large cardinal axioms or working in the Solovay model [21]. Moreprecisely, all our results will be deduced assuming the following coveringproperty for every definable set of reals X :

(CCP) For every collection F of closed sets of reals, either X can becovered by countably many members of F or X contains a G5-Setwhich cannot be covered by countably many members of F .

Some explanations are in order here. The word 'definable' is to be interpreted in its most generous form: there is a formula 4>(v), with some parameters which are either reals or ordinals, such that X = {x E lR : 4>(x)} .

These kind of covering properties have long history in analysis and descriptive set-theory. The paper [19] of Solecki is perhaps most up-to-date,giving an excellent survey of the previous work as well as some new results.

1Research partially supported by NSERC and SFS.

213

CA. Di Prisco et al. (ed.), Set Theory, 213-226.© 1998 KluwerAcademic Publishers.

214 STEVO TODORCEVIC

For example, Solecki [19] gives a proof of CCP for analytic sets and presentsa proof that CCP is true for every definable set using the existence of ameasurable cardinal which dominates infinitely many Woodin cardinals.This proof, due to D.A. Martin, follows rather closely the proof of AD fordefinable sets of reals (see [14], [15], [27]).

The CCP in the Solovay model has been established in [2]. Solovay'smodel M = V Coll(w,<5) cannot be ignored here because under suitablelarge cardinal assumptions, L(lR.) is elementarily embeddable into L(lR.)M,and because M, moreover, shows that the consistency strength of CCP isway below the consistency strength of AD. We should also note that the factthat in Solovay's model CCP is true under some considerable restriction onF (i.e., F = {Fx : x E NN} for some closed F ~ (NN)2) was also firstobserved in [19] using a result of Louveau [12].

After a short survey of known results, we start our presentation of applications of CCP by showing how to use CCP to analyze gaps in P(N)/ fin.We assume the reader is familiar with [23] and [24], but let us recall someof the basic definitions and notations from these two papers. The power-setalgebra P(N) of sets of integers is equipped with the standard compactmetric topology obtained from its identification with the Cantor-cube 2N•

For subsets a and b of N, let a .1 b denote the fact that their intersectionis finite. For A,B ~ P(N) , let A .1 B denote the fact that a.l b whenevera E A and b E B. Let A1. denote the set of all b ~ N such that b .1 a for alla E A . A subset c of N separates A and B if c J.. A and (N\ c) J.. B . We shallsay that A, B ~ P(N) form a gap if A .1 B and there is no c ~ N whichseparates them. We shall also say that A is countably generated in someD ~ P(N) if there is a countable C ~ D such that for every a E A there isc E C such that a \ c is finite. An ideal I on N will always be assumed tobe proper i.e., such that N f/: I and that it includes the ideal Fin of finitesubsets of N. We shall say that I is a P-ideal if for every sequence {an} ofelements of I there is b E I such that an \ b E Fin for all n.

Theorem 1. Suppose A and B are two orthogonal families and that Ais definable. Then A is countably generated in BJ. if and only if everycountable subset of B can be separated from A.

Proof Let F = {P(c) : c E BJ.}. Note that the fact that A is not countablygenerated in BJ. is the same as the fact that no countable subfamily of Fcovers A. Thus if A is not countably generated in BJ., applying CCP thereis a G5-set G included in A which is also not countably generated in BJ.. ByTheorem 1 of [23] applied to G and B we conclude that there is a countablesubset of B which cannot be separated from G, and therefore, from A. Thisfinishes the proof. 0

DEFINABLE IDEALS AND GAPS IN THEIR QUOTIENTS 215

Theorem 2. The orthogonal of every definable P-ideal of subsets of N iscountably generated.

Proof. Let I be a given definable P-ideal on N. Then its orthogonal I.Lis also definable and so Theorem 1 is applicable to A = I.L and B = I.Since I is a P-ideal every countable subset of B can be separated from Aso we conclude that A is countably generated in B.L = A. This finishes theproo£ 0

Remark 1. Theorem 2 for analytic ideals can be proved without usingCCP or any other additional set-theoretical assumption using a slightlydifferent route of reasoning which in particular uses the analytic analogueof Theorem 5 below.

Corollary 3. Suppose A and B are two orthogonal P-ideals. If one of themis definable, then they can be separated.

Remark 2. Note that this shows that if A and B are the P-ideals generated by the two sides of Hausdorff's (WI, wI)-gap then neither of them isdefinable. Theorem 2 hints that definable P-ideals on N are in some sensesimple. This turns out to be a correct intuition as the following fact shows.

Theorem 4. Every definable P-ideal I on N must in fact be analytic.

Proof. Let M(I) be the family of all closed approximations to I (see [8])i.e., the family of all compact subsets K of 2N such that

(a) K is monotone, i.e, as; b E K implies a E K(b) for every a E I there is n such that a \ {I, . .. ,n} E K.Let M be the space of all compact monotone subsets of the cantor setviewed as a compact metric space itself when equipped with the exponentialtopology. Note that the proof of Theorem 1 show that in fact we have astrong form of Hurewicz phenomenon here: either A is countably generatedin B.L or there is a B-tree all of whose branches are in A (see [23; Theorem3]). This in particular gives (see [23; p. 59]) that a subset X of a compactmetric space E is E; if and only if for every countable D S; E \ X there is aGc5-set G 2 D disjoint from X. Applying this for E = M and X = M(I),in order to show that M (I) is Fu in M it suffices to show that everycountable 1) S; M \ M(I) can be separated from M(I) by a Gc5-set. Foreach K E 1) pick aK E I such that «« \ {I, .. . ,n} ¢ K for all n . Let a E Ibe such that o« is almost included in a for all K E 1). Let

9 = {K EM : a \ {I, .. . ,n} ¢ K for all n EN}.

Then 9 is a Gc5 subset of M which contains 1) and is disjoint from M(I).To show that the filter M (I) of closed approximations to I is countably


generated it suffices now to show (assuming that I itself is non-atomic)that for every compact set C ~ M (I) there is a non-empty relatively clopensubset U of C such that nU E M(I) . So let {Un} enumerate all relativelyclopen subsets of C. For x E I and n E N, set

Cn(x) = {KEC: x\{l, ... ,n}EK}

Then C = U~=l C(x) and each Cn(x) is closed so by the Baire CategoryTheorem there exist (n(x),m(x)) E N2 such that Um(x) ~ Cn(x)(x). Pick(n, m) such that

I(n,m) = {x E I : (n(x), m(x)) = (n, m)}

is cofinal in I in the ordering of almost inclusion. It follows that K = nUm

contains x \ {I, . . . , n} for all x E I(n,m) and so it belongs to M(I).Let {Km } be a decreasing sequence of elements of M(I) which generates

it. The proof of the theorem is finished once we show that a subset x of Nbelongs to I if and only if for all m there is n such that x \ {I, . . . , n} E K m .

To show the non-trivial (converse) implication consider an x rf. I. ApplyingCCP to the restriction of I to x and the family of all closed nowhere-densesubsets of the cantor cube 2x we conclude that Inp(x) has the property ofBaire in this space (see [2]) . By a result of Jalali-Naini [7] and Talagrand [22]we conclude that there is a partition of x into disjoint finite sets Xi suchthat UiEI Xi rf. I for all infinite I ~ N. Let

K = {y ~ N : Xi ~ Y for all i}.

Then K E M(I) and x \ {I, .. . , n} rf. K for all n. Choose an mEN suchthat K m ~ K . It follows that x \ {I, . .. , n} rf. K m for all n E N. Thisfinishes the proof. 0

Remark 3. The fact that every analytic P-ideal is Fuo is originally dueto S. Solecki [20] who at the crucial step uses a version of a theorem ofChristensen and Saint-Raymond (see [10]).

We shall now see that a slightly faster sequence of generators of M (I)together with some ideas of Bartoszynski and Fremlin (see [1], [4]) give usa mapping witnessing the following interesting fact.

Theorem 5. If I is a definable P-ideal then there is a monotonic mapfrom the Banach lattice £1 onto a cofinal subset of I .

Proof. First of all note that for every closed approximation K to I there isa closed approximation H to I such that

Hll.H = {xUy: x,yEH}


is included in K (see [20]) . Thus, the sequence {Kn } which generates M(I)can be refined to a subsequence {Gn } such that

(1)

(3)

(2)

One use of so fast a sequence is in proving that whenever we take a sequence{vn} of finite sets such that Vn E Gn for all n then the union U~=1 vn belongsto I. For x E I, fix a strictly increasing sequence of integers {kn(x)} suchthat

x \ {I, .. . , kn(x)} E Gn+2

for all n. Choose F : I ~ £1 (N x Fin) so that

F(x)(kn(x), xn{I, ... ,kn(x)}) _ 2- n

for all x E I and n E N. For 0" E £1(N x Fin) set

I q = {XEI : F(x):S;O"}

and define cI> : £1 ~ P(N) by cI>(0") = UIq • Note that cI> is monotonic andthat

cI>(F(x)) 2 x,

so the proof is finished once we show that cI> (0") E I for all 0" E £1 (Nx Fin) .Fix such 0", and for nEZ, set

Dn = {(k, t) EN x Fin : O"(k, t) ~ 2-n}

Then there must be no such that IDnl :s; 2nfor all n ~ no. For n ~ no, letVn be the union of all sets of the form t \ {l, ... , f} for which there exist8 E Fin and kEN such that

k > E, (k,t) E Dn+! and (£,8) E ti;

8 = t n {I, . .. ,£}

t \ {I, .. . , £} E Gn+2'

(4)

(5)

(6)

Then by (1) we have Vn E Gn+! for all n ~ no and so by the remark aboveu = U~=no Vn belongs to I. Fix an integer £0 above all integers belongingto the first coordinate of a pair from Dno' Note that for every x E I q andn E N, O"(kn(x), X n {I, ,kn(x)}) ~ 2-n, so in particular

xn{kn(x)+I, , kn+!(x )} ~ Vn forall n~no. (7)

It follows that cI>(0") = UIq ~ uU{I , . . . , Eo}. This completes the proof. 0


The analytic analogue of the following companion result to Theorem 5was known even before the analytic analogue of Theorem 5 itself (see [23;Theorem 6]).

Theorem 6. Let I be a definable P-ideal on N which is not generatedby a single subset of N and Fin . Then there is a monotonic map whichtransfers I to a cofinal subset of NN.

Proof. Suppose first that there is a sequence {en} of elements of I.L whichcannot be separated from I. Define ~ : I ~ NN by

~(a)(n) = min{m: anCi ~ {1, .. . ,m} for all i $ n} .

Clearly ~ is monotone and every ~(a) is a monotonic function of NN.Note also that if for some A ~ I the image ~"A is bounded in NN underthe ordering of eventual dominance then A can be separated from {en}.Applying CCP to I and the family P(b)(b E {en}.L) we conclude that thereis a Go-set G ~ I which cannot be separated from {en} and therefore itsimage ~"G is unbounded in NN. Let Io be the ideal generated by G andFin . Then Io is an analytic ideal and Io and {en} satisfy the hypothesis ofTheorem 5 of [23] so the image ~"Io, and therefore, ~III is cofinal in NNin the ordering of eventual dominance . As in [23; p. 63] we find a kEN sothat 'li : I ~ NN given by

'li(a)(n) = ~(a)(n + k)

has a cofinal range even in the ordering of everywhere dominance.This finishes the proof modulo proving that the assumption about the

existence of {en} can be made. To get into that situation we proceed as inthe proof of Theorem 6 of [23] and transfer I to another definable P-idealt. generated by {a : a E I}, where

a = {2i(2j+1): iEN, j$lan{l, ... ,i}I}.

Since a t-7 a is a monotonic map it suffices to prove the theorem for i inplace of I . One of the reasons for moving to i is that it cannot be separatedfrom its orthogonal (see [23; p. 64]). Applying Theorem 1 (or 2) to i and itsorthogonal we conclude that the only possible alternative is that there is acountable subset of the orthogonal of i which cannot be separated from iwhich is exactly what we are looking for. This completes the proof. 0

The last two results are giving us some information about the followingpreordering between ideals on the integers: I $ .J if and only if there is amonotonic map

~:.J~I

which maps .:1 onto a cofinal subset of I . It is not hard to see that this isequivalent to saying that .:1 is 'cofinally finer' in the sense of Tukey [25] i.e.,there is a Moore-Smith convergent map from .:1 into I. The fact I ~ .:1 issometimes expressed also with the words 'I is Tukey-reducible to .:1'. Tukeyordering has received a considerable attention in recent times especiallyafter the beautiful paper of Fremlin [4] which has synthesized most of theprevious work of Bartoszynski, Cichon, Miller, Raisonnier , Stern, Truss andothers about the cardinal invariants of the continuum (see also [1], [26]) .

Definable ideals on N tend to represent , in the Tukey sense, almost anynatural notion of smallness that one can find in other areas of mathematics.For example, £1 is a representation of the ideal of Lebesgue measure zerosets of reals. In fact , £1 with the ordering of eventual dominance is Tukeyequivalent to the Lebesgue measure zero ideal. Note also that NN representsthe ideal of compact subsets of the irrationals. The ideal K(Q) of compactsubsets of the rationals is another interesting ideal. In particular, it is anexample of an ideal I such that I ~ £1 (see [5]). The fact that the idealof nowhere-dense subsets of Q is Tukey-reducible to £1 is one of the finestresults of the theory (see [4; Theorem 3B] and [1; §2]) .

It should be mentioned that the program of finding Tukey-reductions(or non-reductions) between some standard ideals like NN, £1, NWD, Zo wasinitiated long ago by J . Isbell [6], who proved, for example, that DN == Dfor each of these four directed sets. (In connection with this, note that Theorem 6 immediately gives IN == I for every non-atomic definable P-idealI on N.) Proving non-reducibility can sometimes be quite non-trivial sincewe do not require that the connecting maps be definable. For example, it isonly recently that Louveau and Velickovic [13] showed that £1 ~ Zoo(HereZo denotes the ideal of subsets of N of asymptotic density zero.) An earlyresult of that sort, essentially appearing already in Isbell's paper [6], statesthat Zo ~ NWD is also quite non-trivial (see also [4; p. 211]). Many relations between these standard ideals, however, still remain unsettled. Thereader is referred to Fremlin's problem list for some of the most attractive questions of this sort, especially those which would further clarify therelationship between NWD and £1 .

Let us now turn our attention to the classical phenomenon of Hausdorff's gaps inside the quotient algebras P(N)jI over definable ideals I onN. It turns out that one can say much more than just to assert the existence of a Hausdorff gap in P(N) [T. The first general fact is that for everydefinable ideal I on N there is an embedding

 : P(N)j fin -7 P(N)jI

with a lifting <p.: P(N) -+ P(N) which is special in many ways, andin particular, it is continuous . To see this, recall that CCP implies thatevery set of reals has the property of Baire (see [2]), so in particular I isBaire-measurable. By a result of Jalali-Naini [7] and Talagrand [22] thereis a partition

00

N = UFii=l

into disjoint finite sets Fi such that every infinite union of Fi'S is outsideI. Then

<P.(x) = UFiiEx

(8)

is a continuous (and more) lifting of an embedding of P (N) / fin intoP(N)/I. It turns out that every such embedding transfers Hausdorff's gapsof P(N)/ fin into Hausdorff's gaps of P(N)/I, a result which in the casewhen I is analytic was first proved in [24].

Theorem 7. If I is a definable ideal on N, then every embedding <P ofP(N)/ fin into P(N)/I with a continuous (equivalently Baire-measurable)lifting preserves all Hausdorff's gaps.

Proof. Let <p. : P(N) -+ P(N) be a fixed continuous lifting of the embedding <.P and let (A, B) be a given Hausdorff gap of P(N)/ fin i.e., twoorthogonal P-ideals on N which cannot be separated. Suppose its image(<p~A, <P~B) is not a gap in P(N)/I. This means that there is a c ~ N suchthat

Let

<p. (a) \ c E I for all a E A

<p.(b) n c E I for all bE B

(9)

(10)

A. = {a ~ N: e, (a) \ c E I}

Then A. is a definable and it includes A. Consider an a E A. and b E B .Then

<.p.(a) n <.p.(b) ~ (<P.(a) \ c) U (<p.(b) n c) E I (11)

Since <P.(a n b) = <P.(a) n <p.(b) (modI), we conclude that <P.(a n b) E I .Since <p. is a lifting of an embedding of P(N) / fin into P(N) /I this can onlyhappen if a n b is finite. This shows that A. and B are orthogonal to each

other. Since B is a P-ideal, every countable subset of B can be separatedfrom A. so, by Theorem 1, we conclude that there is a sequence {en} ofelements of B.l.. which generate A. and therefore A. Thus, every elementof A is almost included in some en. Since A is also a P-ideal there mustbe n so that en almost includes every element of A, and in particular, itseparates A and B, a contradiction. This finishes the proof. 0

Corollary 8. If I is a definable ideal on N, then the quotient algebraP(N)jI contains an (Wl ,wn-gap.

It turns out that in general embeddings of P(N)j fin into P(N)jI withcontinuous lifting do not necessarily preserve all gaps of P(N)j fin i.e., thatin Theorem 7 restriction to Hausdorff gaps was necessary. Example 1 of[24] describes a situation where the non-preservation of gaps happens.

Theorem 9. Let I be a definable P-ideal on N. Then every embedding of P(N) j fin into P(N) jI which has a continuous lifting preserves all gaps.

Proof. This follows from Theorem 4 and Theorem 11 of [24] but for theconvenience of the reader we sketch the proof of the special case when is obtained via a lifting as defined above in (8). One of the rather strongproperties of this . is that it is completely additive i.e,

.(A) = U.(Ai )

iEI

(12)

for every disjoint family Ai (i E I) of subsets of N whose union is equal to A(moreover, .(0) = 0 and .(N) = N) . Note that every completely additivelifting is a continuous function from P(N) into P(N). So, let (A, B) be agiven gap of P(N)jfin and suppose its image (~A,~B) is not a gap inP(N)jI i.e., that there is c ~ N such that

.(a) \ c E I for all a E A, and

.(b) n c E I for all bE B.

As before, we consider the sets

A. = {a~N : .(a)\cEI}, andB. = {b ~ N : .(b) nc E I}.

(13)

(14)

Then as before one checks that A. and B. are two definable orthogonalideals on N which contain A and B respectively. So, in particular A. andB. form a gap in P(N)j fin. The following Claim together with Corollary 3give us the desired contradiction.


Claim. A* and B* are P-ideals.

Proof. By symmetry it suffices to check this for A* , so let A o be a givencountable subset of A* . Applying the fact that I is a P-ideal to (13), weget an x E I such that

(ep*(a) \ c) \ x is finite for all a E Ao.

By (12), this means that for every a E Ao there is ka EN such that

ep*(a \ {I, .. . ,ka } ) \ c ~ x,

Setd = U (a \ {I, ... ,ka } ) .

aEAo

(15)

(16)

Clearly every element of Ao is almost included in d so it suffices to showthat d belongs to A*. To see this, note that (using again (12)):

ep*(d)\c = U ep*(a\{I, ... ,ka})\c~x. (17)aEAo

This completes the proof of the Claim. D

Now Theorem 9 follows from the Claim. D

Corollary 10. The quotient algebra P(N)jI over any definable P-idealcontains a (b, w*)-gap.

Any study of homomorphisms ep : P(N)jI -t P(N).J between quotients over definable ideals I and .J always reduces to the study of thecorresponding liftings ep* : P(N) -t P(N) related to them via the commutative diagram

P(N)

P(N)jI

P(N)

P(N)j.J

where 11"0 and 11"1 are the corresponding quotient maps. Key results of thissubject tend always to be assertions that certain kind of liftings can be

improved to better ones . For example, we needed such a result in the proofof Theorem 9 above since the proof when the lifting <l). is only continuousrather than completely additive would be considerably more difficult. Thefollowing are two general problems in this new theory of liftings.

Problem 1. Describe the class of definable ideals I which have the property that every homomorphism <l) from P{N)/ fin into P{N)/I with a Bairemeasurable lifting has also a completely additive lifting <l). : P{N) -+ P{N).

Problem 2. Describe the class of definable ideals I with the property thatevery homomorphism <l) from P{N)/ fin into P{N)/I has an additive lifting,i.e., a lifting <l). : P{N) -+ P{N) which itself is a homomorphism.

Of course, it also makes sense considering the versions of the problemwhen the homomorphism <l) is further required to be onto or to make othernatural restrictions. A deep result concerning Problem 1 was recently obtained by 1. Farah [3] who has isolated a natural class of 'non-pathologicalanalytic P-ideals' (including virtually every known example of analytic Pideal) for which Problem 1 has a positive answer.

Note that every completely additive lifting of a homomorphism <l) fromP{N)/fin into P{N)/I determines (and is determined by) a Rudin-Keislerreductionof I to Fin, i.e., a map h : N -+ N such that the preimage h-1{a)of a set a ~ N belongs to I if and only if the set a is finite. (If h is moreoverfinite-to-one then we say that h is a Rudin-Blass reduction.) The first resultof this sort is due to Mathias [16] who proved that every analytic ideal I isRudin-Blass reducible to Fin using the famous proof involving the Ramseyproperty of analytic sets. This result was later extended by Jalali-Naini [7]and Talagrand [22] who showed that the relation Fin ~RB I is, in fact , acharacterization of the class of Baire-rneasurable ideals on N.

In [20], Solecki has obtained an interesting analogue of Mathias' Theorem saying that an analytic ideal I is Rudin-Blass reducible to Fin x 0 (theideal of subsets of N2 whose projections on the first coordinate are finite)if and only if I is not a P-ideal. The following result (whose proof needsonly the consequence of OOP which says that definable sets of reals areBaire-measurable) is an analogue of Solecki's theorem for definable idealsbut its proof is more direct than the one appearing in [20].

Theorem 11. A definable ideal I on N is Rudin-Blass reducible to Fin x 0if and only if it is not a P-ideal.

Proof. To see the non-trivial direction, assume I is not a P-ideal. Movingfrom the index-set N to N2 this means that we are working with an ideal Ion N2 such that every vertical section {i} x N belongs to I but no memberof I almost includes all of them. Let 0 x Fin be the ideal on N2 orthogonal


to Fin x 0, i.e., the ideal of subsets of N2 which have finite intersectionwith every vertical section {i} x N. (We have already encountered this idealin Theorem 6 above in a slightly different guise.) Let Io be the ideal on N2

generated by I and 0 x Fin. Note that by our assumption on I, the idealIo is proper. Set

J = {a~N: NxaEIo} .

Then J is also proper, definable and, therefore, Baire-measurable. By theresult of Jalali-Naini and Talagrand we can choose a partition of N intodisjoint finite intervals Xk (k E N) such that UkEf Xk is outside of J forevery infinite set I ~ N. Let ¢ : N2 -+ N be a fixed bijection. Defineh: N2 -+ N2 by

h(p, q) = (i,j) iff p ~ i and q E X4>(i,j)'

Clearly h is finite-to-one . To see that h is a Rudin-Blass reduction of Ito Fin x 0 consider a b ~ N2 . If b E Fin x 0 its projection to the firstcoordinate is bounded by some integer m so the preimage h-1 (b) will haveits projection on the first coordinate bounded by the minimal integer kwhich has the property that ¢(k,£) > m for all £. Thus, h-1(b) is coveredby finitely many vertical sections which are all in I, and therefore .itselfbelongs to I. On other hand, if b is not in Fin x 0, we can assume thatit is equal to the graph of a function e: J -+ N where J is some infinitesubset of No Let I be the image of b under ¢. Then I is an infinite subsetof N, so the union

x = UXk

kef

is not in J. Let a = N x x . Then a ¢ Io, so in order to show h-1(b) ¢ I itsuffices to show that the set a \ h-1 (b) belongs to 0 x Fin, i.e., it has finiteintersection with every vertical section {p} x N. To see this, let np be theinteger which dominates all sets of the form

X4>(i,e(i» (i E In{I, oo . ,p}).

Consider a (p,q) in a n ({p} x N) with q > np, and pick k E I such thatq E Xk . Then k = ¢(i,j) for some (i,j) E b with i > p. Applying thedefinition of h to the pair (p,q) we conclude that h(p,q) = (i,j) . Thisshows that any pair from a n ({p} x N) with the second coordinate abovenp belongs to h-1 (b) and completes the proof of Theorem 11. 0

Remark 4. Note that Theorem 11 suggests the following natural question:Is Fin x 0 ~RB I true for every Baire-measurable ideal I which is not a


P-ideal? At the moment we don't know the answer to this question, sinceto conclude that the ideal J above is Baire-measurable we seem to needthe assumption that I is definable rather than than just that it is Bairemeasurable.

The quotient P{N)j Fin X 0 is an interesting algebra whose importancehas been recognized very early, and Theorem 11 reveals just another ofits remarkable features . If we are to continue the study of the analogiesbetween P{N)j Fin and P{N)j Fin x 0, the following problem would bequite natural.

Problem 3. Investigate the preservation of gaps under embeddings

q> : P{N)j Fin x 0 -+ P{N)jI

which have Baire-measurable liftings, and in particular, determine the gapspectrum of the quotient P{N)j Fin x 0.

Note that any such definable ideal I will not be a P-ideal as the followinganalogue of Theorem 8 of [24] shows.

Theorem 12. Suppose that a quotient algebra P{N)jI is embeddable intoa quotient algebra P{N)j:r via an embedding which has a Baire-measurablelifting. If:r is a definable P-ideal then so is I .

Proof. This follows from Theorem 4 and [24; Theorem 8]. oWe have seen above (Theorem 4) that the principle of closed covering,

CCP, reduces the class of all definable P-ideals to analytic ones. The structure of analytic P-ideals has been considerably determined in the recentbeautiful paper of S. Solecki [20] who showed, for example, that an analytic P-ideal is either an Fq-ideal or is Rudin-Blass reducible to 0 x Fin .(In this connection recall Theorem 6 above which says that 0 x Fin is alsoa critical ideal when we consider the ordering of Tukey-reducibility.) So byTheorem 4 this dichotomy is also true for all definable P-ideals (assumingagain, of course, CCP). It follows that the quotient algebra P(N)j0 x Fin isembeddable into any quotient P(N)jI over a definable P-ideal which is notFq via an embedding with a completely additive lifting. So the followingproblem is now quite natural.

Problem 4. Investigate the preservation of gaps under embeddings

e :P{N)j0 x Fin -+ P{N)jI

which have Baire-measurable liftings, and in particular, determine the gapspectrum of the quotient algebra P(N) /0 x Fin.


The remaining class of quotients over Fa P-ideals might also have somecritical elements but this has not yet been established (see [9], [17]). Itshould also be noted that the investigation of the structure of Fa idealsthemselves is also a well established subject and for this the reader is referred to the article of C. Laflamme [11].

References

1. T. Bartoszynski, Measure-category invariants, preprint 1997.2. C. Di Prisco and S. Todorcevic, Perfect-set properties in L(R), preprint 1997.3. I. Farah, Analytic ideals and their quotients, Ph.D . Thesis, University of Toronto,

1997.4. D. Fremlin, The partially ordered sets of measure theory and Tukey's ordering , Note

di Matematica vol. 11 (1991), 177-214.5. D. Fremlin, Families of compact sets and Tukey's ordering, Atti. Sem. Mat . Fis.

Univ. Modena 39 (1991), 29-50.6. J .R. Isbell , Seven cofinal types, J. London Math. Soc. 4 (1972), 651-654.7. S.-A. Jalali -Naini, The monotone subsets of Cantor space, filters and descriptive set

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American Mathematical Society, 2 (1989), 71-125.16. A.R.D. Mathias, A remark on rare filters, in: Coli. Math. Soc. Janos Bolyai 10,

Infinite and Finite sets (A. Hajnal et al. eds) North Holland 1975.17. K. Mazur, Towards a dichotomy for FCT-ideals, preprint 1996.18. Gy. Petruska, On Borel sets with small cover: a problem of M. Laczkovich, Real

analysis exchange, 18 (2) (1992-3), 330-338.19. S. Solecki, Covering analytic sets by families of closed sets, J. Symbolic logic, 59

(1994), 1022-1031.20. S. Solecki, Analytic ideals, Bulletin of Symb . Logic 2 (1996), 339-348.21. R.M. Solovay, A model of set theory in which every set of reals is Lebesgue mea

surable, Ann of Math. (2) 92 (1970), 1-56.22. M. Talagrand, Compacts de fonctions measurables et filters nonmeasurables, Studia

Math . 67 (1980) , 13-43.23. S. Todorcevic, Analytic gaps, Fund. Math. 150 (1996), 55-66.24. S. Todorcevic, Gaps in analytic quotients, preprint 1996.25. J .W. Tukey, Convergence and uniformity in topology, Ann. Math. Studies 1, Pr ince

ton Univ. Press, 1940.26. P. Vojtas , Generalized Galois-Tukey connectives between explicit relations on clas

sical objects of real analysis, Israel Math. Conference Proceedings vol. 6 (1993),619-643.

27. W.H. Woodin , Supercompact cardinals, sets of reals, and weakly homogeneous trees,Proc. Natl. Acad. Sci . USA. 85 (1988), 6587-6591.

set theory: techniques and applications cura§ao 1995 and barcelona 1996 conferences

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