saharon shelah- diamonds

8/3/2019 Saharon Shelah- Diamonds

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DIAMONDS

SH922

SAHARON SHELAH

Abstract. If = + = 2 > 1 then diamond on holds. Moreover, if = + = 2 and S { < : cf() = cf()} is stationary then S holds.Earlier this was known only under additional assumptions on and/or S.

Date: Oct.7, 2009.Research supported by the United-States-Israel Binational Science Foundation (Grant No.

2002323), Publication No. 922. The author thanks Alice Leonhardt for the beautiful typing.First Typed - 07/Sept/19 Previous Version - 09/Oct/5.

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2 SAHARON SHELAH

1. Introduction

We prove in this paper several results about diamonds. Let us recall the basicdefinitions and sketch the (pretty long) history of the related questions.

The diamond principle was formulated by Jensen, who proved that it holds in Lfor every regular uncountable cardinal and stationary S . This is a predictionprinciple, which asserts the following:

{0z.1}Definition 1.1. S (the set version).

Assume = cf() > 0 and S is stationary, S holds when there is asequence A : S such that A for every S and the set { S :A = A} is a stationary subset of for every A .

The diamond sequence A : S guesses enough (i.e., stationarily many)initial segments of every A . Several variants of this principle were formulated,for example:

{0z.4} Definition 1.2. S.Assume = cf() > 0 and S is a stationary subset of . Now

S holds when

there is a sequence A : S such that each A is a subfamily ofP(), |A| ||and for every A there exists a club C such that A A for every C S.

We know that S holds in L for every regular uncountable and stationaryS . Kunen proved that S S. Moreover, if S1 S2 are stationary subsetsof then S2

S1

(hence S1). But the assumption V = L is heavy. Tryingto avoid it, we can walk in several directions. On weaker relatives see [Sh:829] andreferences there. We can also use other methods, aiming to prove the diamondwithout assuming V = L.

There is another formulation of the diamond principle, phrased via functions(instead of sets). Since we use this version in our proof, we introduce the following:

{d.21}Definition 1.3. S (the functional version).

Assume = cf() > 0, S , S is stationary. S holds if there exists adiamond sequence g : S which means that g for every S, and forevery g the set { S : g = g} is a stationary subset of .

By Gregory [Gre76] and Shelah [Sh:108] we know that assuming = + = 2

and = cf() = cf(), < , and GCH holds (or actually just = or ( and > 0

is strong limit, then (by [Sh:460]) the set { < : S fails} is bounded in (recall that is regular). Note that the result here does not completely subsumethe earlier results when = 2 = + as we get diamond on every stationary setS \Scf(), but not

S; this is inherent as noted in 3.4. In [Sh:829], a similar,

stronger result is proved for S : for every = + = 2 > , strong limit for

some finite d Reg , for every regular < not from d we have S , and even

S for most stationary S S . In fact, for the relevant good stationary setsS S we get

S . Also weaker related results are proved there for other regular

(mainly = cf(2)).


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The present work does not resolve:{0z.7}

Problem 1.4. Assume is singular, = + = 2, do we have Scf()

? (You may

even assume G.C.H.).However, the full analog result for 1.4 consistently fails, see [Sh:186] or [Sh:667];

that is: if G.C.H., > cf() = then we can force a non-reflecting stationary

S S+

such that the diamond on S fails and cardinalities and cofinalities arepreserved; also G.C.H. continue to hold. But if is strong limit, = + = 2, stillwe know something on guessing equalities for every stationary S S ; see [Sh:667].

Note that this S (by [Sh:186], [Sh:667]) in some circumstances has to be small

() if ( is singular, 2 = + = , = cf() and) we have the square (i.e.there exists a sequence C : < , is a limit ordinal, so that C is closedand unbounded in , cf() < |C| < and if is a limit point of

C then C = C ) then S holds, moreover, if S S reflects in a

stationary set of < then S holds, see [Sh:186, 3].

Also note that our results are of the form S for every stationary S S for

suitable S . Now usually this was deduced from the stronger statement S.However, the results on S cannot be improved, see 3.4.

Also if is regular we cannot improve the result to S , see [Sh:186] or [Sh:587],

even assuming G.C.H. Furthermore the question on 2 when 21 = 2 = 2

0 wasraised. Concerning this we show in 3.2 that

S21

may fail (this works in other

cases, too).{0z.11}

Question 1.5. Can we deduce any ZFC result on strongly inaccessible?By Dzamonja-Shelah [DjSh:545] we know that failure of SNR helps (SNR stands

for strong non-reflection), a parallel here is 2.3(2).For =


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4 SAHARON SHELAH

2. Diamond on successor cardinals

Recall (needed only for part (2) of the Theorem 2.3):{ya.1}

Definition 2.1. 1) We say on S has -SNR or SNR(,S,) or has strong non-reflection for S in when S S := { < : cf() = } so = cf() > = cf()and there are h : and a club Eof such that for every SE for some clubC of, the function h C is one-to-one and even increasing; (note that without lossof generality nacc(E) successor and without loss of generality E = , so Reg \+ SNR(, S , )). If S = S we may omit it.

{ya.2}Remark 2.2. Note that by Fodors lemma if cf() = > 0 and h is a functionfrom some set and the range of h is then the following conditions areequivalent:

(a) h is one-to-one on some club of

(b) h is increasing on some club of (c) Rang(hS) is unbounded in for every stationary subset S of .

Our main theorem is:{ya.4}

Claim 2.3. Assume = 2 = +.1) If S is stationary and S cf() = cf() then S holds.2) If0 < = cf() < andS fails where S = S

(or just S S

is a stationary

subset of ) then we have SNR(, ) or just has strong non-reflection for S Sin .

{ya.6}

Definition 2.4. 1) For a filter D on a set I let Dom(D) := I and S is calledD-positive when S I (I\S) / D and D+ = {S Dom(D) : S is D-positive}and we let D + A = {B I : B (I\A) D} (so if D = D, the club filter on the

regular uncountable then D+ is the family of stationary subsets of X).2) For D a filter on a regular uncountable cardinal which extends the club filter,let D means: there is f = f : S which is a diamond sequence for D(or a D-diamond sequence) which means that S D+ and for every g theset { < : g = f} belongs to D+; so f is also a diamond sequence for thefilter D + S, (clearly S means D+S for S a stationary subset of the regularuncountable ).

A somewhat more general version of the theorem is{ya.5}

Claim 2.5. 1) Assume = + = 2 and D is a -complete filter on whichextends the club filter. If S D+ and S cf() = cf() then we have D+S.2) We have

Dwhen :

(a) =


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DIAMONDS SH922 5

3) Assume = + = 2 and 0 < = cf() < , S S is stationary, and D is a

-complete filter extending the club filter on to which S belongs. If D fails then

SNR(,S,).

Proof. Proof of 2.3 Part (1) follows from 2.5(1) for D the filter D + S. Part (2)follows from 2.5(3) for D the filter D + S. 2.3

Proof. Proof of 2.5Proof of part (1)

Clearly we can assume

0 > 0 as for = 0 the statement is empty.

Let

1 f : < list the set {f : f is a function from to for some < }.

For each < clearly || so let

2, u, : < be -increasing continuous with union such that < |u,| 0 + || < .

For g let hg be defined by

3,g hg() = Min{ < : g = f}.

Let cd, cd : < be such that

4 (a) cd is a one-to-one function from onto such that

cd() sup{ : < } (when = : < )(b) for < , cd is a function from to such that

= : < cd(cd()) =

(they exist as = , in the present case this holds as 2 = + = ).Now we let (for < , < ):

5 f1, be the function from Dom(f) into defined by f

1,() = cd(f())

so Dom(f1,) = Dom(f).

Without loss of generality

6 S is a limit ordinal.

For g and < we let

Sg =

S : = sup{ u, : for some u,

we have g = f1,}

.

Next we shall show

7 for some () < for every g the set S

()g is a D-positive subset of .


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6 SAHARON SHELAH

Proof of7 Assume this fails, so for every < there is g such that Sg

is not D-positive and let E be a member of D disjoint to Sg

. Define g by

g() := cd(g() : < ) and let hg be as in 3,g, i.e. hg() = Min{ :g = f}.

Let E = { < : is a limit ordinal such that < hg() < }, clearly itis a club of hence it belongs to D and so E = {E : < } E belongs to Das D is -complete and + 1 < .

As S is a D-positive subset of there is E S. For each < as E E clearly hg() < and as well as hg() belong to {u, : sup(Dom(f) Rang(f)) for every < }, so E is a club of hence it belongs to D. By assumption (g) of the claim the set

Sg := { S : = sup{ u : ( u)(f = g )}}is D-positive, so we can choose E E Sg. Hence B := { u : (

u)(f = g)} is an unbounded subset of u and let h : B u be h() =min{ u : f = g}, clearly h is a function from B into u. Now B 0 and let S := S E

.

Clearly v is well defined, see clauses (b),(e) of4, and for S let F = {f22,

,:

v}, so each member is a function from some u into some ordinal < .Let

2 S1 := { S : there are f

, f F which are incompatible as functions}

3 S2 := { S : / S

1 but the function {f : f F} has domain = }

4 S3 = S\(S1 S

2 ).

For S3 let g = {f : f F}, so by the definition of S

: = 1, 2, 3 clearly

g . Now if g : S

3 is a diamond sequence for D then we are done.

So assume that this fails, so for some g and member E of D we have S3 E g

= g . Without loss of generality E is included in E

. But then

we could have chosen (g, E) as (g, E), recalling v was already chosen. Easily thetriple (g, E, v) is as required in 1, contradicting the choice of in 2 so we aredone proving part (2) of Theorem 2.3 hence also part (1). 2.5(2)


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8 SAHARON SHELAH

Proof. Proof of part (3)

We use cd,cd (for < ), u, : < : < , f : < , f1, : sup({ A : f() f()}).

Remark 3.3. Similarly for other cardinals.

Proof. There is an 1-complete 3-c.c. forcing notion P not collapsing cardinals,not changing cofinalities, preserving 2 = for = 0, 1, 2 and |P| = 3 such thatin VP, we have (), in fact more1 than () holds - see [Sh:587]. Let Q be the forcingof adding 2 Cohen or just any c.c.c. forcing notion of cardinality 2 adding 2reals (can be Q

, a P-name). Now we shall show that P Q, equivalently P Q is

as required:

Clause (a):

|P Q| = 3; trivial.Clause (b):

Preserving cardinals and cofinalities; obvious as both P and Q do this.Clause (c): Easy.

Clause (d): In VP we have () as exemplified by say A = A : S. We shall

show that VPQ |= A satisfies (). Otherwise in VPQ we have f = f : Ssay in V[GP, GQ] a counterexample then in V[GP] for some q Q and f

we have

V[GP] |= (q Q f

= f : S where f

: A 1 for each S

form a counterexample to ()).

Now in V[GP] we can define g = g1 : S V[GP] where g1 a function with

domain A, by

g1() = {i : q f() = i}.

1I.e. there is A = A : S where A is an unbounded subset of of order type 1satisfying:

if f = f : S, f (A)1 then there is f (2)1 such that for every S

21

for

every A large enough we have f() = f().


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12 SAHARON SHELAH

So in V[GP] we have q Q S

( A)f() g1()}. Also g

1 () is a countable

subset of 1 as Q satisfies the c.c.c.For S we define a function g : A 1 by letting g() = (sup(g1()) + 1hence g() < 1 so g : S is as required on f in () in V[GP], of course.Apply clause () in V[GP] to g : S so we can find g : 2 1 such thatS

> sup{ A, g() > g()}. Now g is as required also in V[GP][GQ]. 3.2

We may wonder can we strengthen the conclusion of 2.3 to S (of course thedemand in clause (e) and (f) in claim 3.4 below are necessary, i.e. otherwise Sholds). The answer is not as: (the restriction in (e) and in (f) are best possible).

{2b.7}Observation 3.4. Assume = Ord(e) S fails for every stationary subset S of such that

() S S when ( < )[tr = ]

or just

() S ||tr > ||(f) (D)S , see below, fails for every S S when S ||

tr = .

Recalling{2b.9}

Definition 3.5. 1) For = cf() let tr = {|T | : T is closed under

initial segments (i.e. a subtree) such that |T >

| }.2) For regular uncountable and stationary S let (D)S mean that there is asequence P = P : S witnessing it which means:

()P (a) P has cardinality <

(b) for every f the set { S : f P} is stationary

(for successor it is equivalent to S; for strong inaccessible it is trivial).

Proof. Proof of 3.4 Use P = adding +, -Cohen subsets.The proof is straight. 3.4

{2b.11}Remark 3.6. The consistency results in 3.4 are best possible, see [Sh:829].

References

[Gre76] John Gregory, Higher Souslin trees and the generalized continuum hypothesis, Journal ofSymbolic Logic 41 (1976), no. 3, 663671.

[Sh:108] Saharon Shelah, On successors of singular cardinals, Logic Colloquium 78 (Mons, 1978),Stud. Logic Foundations Math, vol. 97, North-Holland, Amsterdam-New York, 1979, pp. 357380.

[Sh:186] , Diamonds, uniformization, The Journal of Symbolic Logic 49 (1984), 10221033.

[Sh:460] , The Generalized Continuum Hypothesis revisited, Israel Journal of Mathematics116 (2000), 285321, math.LO/9809200.


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[DjSh:545] Mirna Dzamonja and Saharon Shelah, Saturated filters at successors of singulars, weakreflection and yet another weak club principle, Annals of Pure and Applied Logic 79 (1996), 289316, math.LO/9601219.

[CDSh:571] James Cummings, Mirna Dzamonja, and Saharon Shelah, A consistency result onweak reflection, Fundamenta Mathematicae 148 (1995), 91100, math.LO/9504221.

[Sh:587] Saharon Shelah, Not collapsing cardinals in (< )support iterations, Israel Journalof Mathematics 136 (2003), 29115, math.LO/9707225.

[Sh:667] , Successor of singulars: combinatorics and not collapsing cardinals in (< )-support iterations, Israel Journal of Mathematics 134 (2003), 127155, math.LO/9808140.

[DjSh:691] Mirna Dzamonja and Saharon Shelah, Weak reflection at the successor of a singularcardinal, Journal of the London Mathematical Society 67 (2003), 115, math.LO/0003118.

[Sh:775] Saharon Shelah, Middle Diamond, Archive for Mathematical Logic 44 (2005), 527560,math.LO/0212249.

[Sh:829] , More on the Revised GCH and the Black Box, Annals of Pure and Applied Logic140 (2006), 133160, math.LO/0406482.

[Sh:898] , pcf and abelian groups, Forum Mathematicum preprint, 0710.0157.

Einstein Institute of Mathematics, Edmond J. Safra Campus, Givat Ram, The He-

brew University of Jerusalem, Jerusalem, 91904, Israel, and, Department of Mathe-

matics, Hill Center - Busch Campus, Rutgers, The State University of New Jersey, 110

Frelinghuysen Road, Piscataway, NJ 08854-8019 USA

E-mail address: [email protected]: http://shelah.logic.at

saharon shelah- diamonds

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