g -dichotomiesg 0-dichotomies andr es eduardo caicedo department of mathematics boise state...

G0-dichotomies

Andres Eduardo Caicedo

Department of MathematicsBoise State University

Logic ColloquiumParis, July 28, 2010

Caicedo G0-dichotomies

This is joint work with Richard Ketchersid.

I want to thank the organizers of the special session for theinvitation to give this talk, and the NSF for partial support throughgrant DMS-0801189.

We work in ZF without choice.

Introduction

Recently, Benjamin Miller established some significant results inclassical descriptive set theory.These results have two components:

Miller proved some new, and extended several known graphtheoretic dichotomies about analytic graphs in Polish spaces,led by the G0-dichotomy of Kechris-Solecki-Todorcevic.His proofs use only “classical” methods, i.e., there are noappeals to forcing or effective descriptive set theory.His techniques are closely tied up to the existence of Suslinrepresentations for the relevant sets. Thus, they generalizefrom the analytic setting to the realm of Suslin sets and, inparticular, hold under ADR of arbitrary graphs on R.

Introduction

Miller observed that, at least in the Borel context, from thegraph theoretic dichotomies, “soft” Baire category argumentsallow one to establish most of the other dichotomy theorems.This includes results of Silver, Harrington-Kechris-Louveau,Harrington-Marker-Shelah, and many others.

Introduction

Ketchersid and I have shown that the appropriate versions of thesegraph theoretic dichotomies hold, for example, in natural models ofAD+, thus obtaining by soft arguments the other dichotomies aswell.Since in general models of AD+ not all sets of reals are Suslin, andthe dichotomies do not seem to reduce to the Suslin case by theusual reflection arguments, an approach different from Miller’s isneeded; we use arguments involving Vopenka-like forcing.

Introduction

Ketchersid and I have shown that the appropriate versions of thesegraph theoretic dichotomies hold, for example, in natural models ofAD+, thus obtaining by soft arguments the other dichotomies aswell.Since in general models of AD+ not all sets of reals are Suslin, andthe dichotomies do not seem to reduce to the Suslin case by theusual reflection arguments, an approach different from Miller’s isneeded; we use arguments involving Vopenka-like forcing.

Suslin sets

Recall that a tree T on X is a subset of X<ω closed under initialsegments. An infinite branch through T is an x ∈ Xω such that forall n,

x � n ∈ T.

We denote by [T ] the collection of all infinite branches through T ,so [T ] 6= ∅ iff T is ill-founded.If y ∈ Y ω and T is a tree on Y × Z, by Ty we mean the tree on Zconsisting of all sequences p ∈ Z<ω such that, letting n = dom(p),there is an x ∈ T with dom(x) = n and

∀i < n(x(i) = (y(i), p(i))

The projection of T is the set

p[T ] = {y ∈ Y ω | [Ty] 6= ∅}.

Suslin sets

x � n ∈ T.

∀i < n(x(i) = (y(i), p(i))

p[T ] = {y ∈ Y ω | [Ty] 6= ∅}.

Suslin sets

x � n ∈ T.

∀i < n(x(i) = (y(i), p(i))

p[T ] = {y ∈ Y ω | [Ty] 6= ∅}.

Suslin sets

x � n ∈ T.

∀i < n(x(i) = (y(i), p(i))

p[T ] = {y ∈ Y ω | [Ty] 6= ∅}.

Suslin sets

Definition.

If κ is an initial ordinal, a set A ⊆ Xω is κ-Suslin iff A = p[T ] forsome tree T on X × κ. A is Suslin iff it is κ-Suslin for some κ. IfA ⊆ ωω is Suslin and κ is least such that A is κ-Suslin, we saythat κ is a Suslin cardinal.

For example, ω and ω1 are Suslin and, under determinacy, the nextSuslin cardinal is ωω.In the context of AD + DC, ADR is equivalent to asserting thatevery set of reals is Suslin.

Suslin sets

Definition.

Suslin sets

Definition.

Hypergraphs and colorings

For d ≤ ω, a d-dimensional hypergraph G on a set X is just asubset of [X]d.Given such G, a Y -coloring of G is a function c : X → Y such that

G(xi | i < d) =⇒ |c[{xi | i < d}]| > 1.

A set A ⊆ X is G-discrete iff [A]d ∩G = ∅. Note that c : X → Yis a Y -coloring of G iff for all y ∈ Y , c−1[{y}] is G-discrete.We will be interested in coloring with ordinals.

G(xi | i < d) =⇒ |c[{xi | i < d}]| > 1.

The graphs G0(d)

For d ≤ ω fix s = (sn | n ∈ ω) dense in d<ω with sn ∈ dn.

Define the hypergraph G0(d) on dω by:

G0(d)(xi | i < d) ⇐⇒∃n∀i, j < d[xi � (n+ 1) = sn

_(i) &∀k > n(xi(k) = xj(k))]

G0(d) is Σ02(s), in fact, D2(Σ0

1(s)). G0(2) is a graph in the usualsense and also denoted G0.

The graphs G0(d)

_(i) &∀k > n(xi(k) = xj(k))]

The graphs G0(d)

_(i) &∀k > n(xi(k) = xj(k))]

Restriction on colorings of G0(d).

Any G0(d)-discrete set A with the property of Baire must bemeager.

The proof is a straightforward Baire category argument.Thus for any Baire-measurable coloring c : dω → Y of G0(d),c−1[{y}] is meager, hence, meager sets can not be closed under|Y |-sized unions.This places limitations on definable colorings, for example, therecan not be a Baire measurable ω-coloring, or (under AD) anycolorings by ordinals.

Kechris-Solecki-Todocevic

G0(d)-dichotomy (d < ω) for analytic graphs:

For G an analytic d-dimensional hypergraph on R exactly one ofthe following hold:

1 G is ω-colorable via a Borel measurable map.

2 There is a continuous map π : dω → R so that π is ahomomorphism of G0(d) into G.

The second possibility will be denoted G0(d) ≤c G.(We already knew that these possibilities are mutually exclusive.)

Kanovei

G0(d)-dichotomy (d < ω) for κ-Suslin graphs:

Let G be a κ-Suslin graph on R, then exactly one of the followinghold:

1 There is a < κ+-Borel measurable κ-coloring of G.

2 G0(d) ≤c G.

Kanovei has also extended the G0-dichotomy to OD(R)-graphs onreals in the Solovay model.

Kanovei

G0(d)-dichotomy (d < ω) for κ-Suslin graphs:

Let G be a κ-Suslin graph on R, then exactly one of the followinghold:

1 There is a < κ+-Borel measurable κ-coloring of G.

2 G0(d) ≤c G.

Kanovei has also extended the G0-dichotomy to OD(R)-graphs onreals in the Solovay model.

Miller

Miller isolated a version of the G0-dichotomy that has as acorollary the Harrington-Kechris-Louveau Theorem. This resultessentially states that |R/E0| is the successor of |R| whenrestricting to appropriately definable relations and reductions.Let {s2n}n be a dense set in 2<ω with s2n ∈ 22n and let {s2n+1}nbe a dense set in 2<ω × 2<ω withs2n+1 = (s2n+1,0, s2n+1,1) ∈ 22n+1 × 22n+1. Now set:

Geven(x, y) ⇐⇒ ∃n∃z ∈ 2ω (x = s2n_(0)_z& y = s2n

_(1)_z),Hodd(x, y) ⇐⇒ ∃n∃z ∈ 2ω (x = s2n+1,0

_(0)_z&y = s2n+1,1

_(1)_z).

Miller

_(1)_z),Hodd(x, y) ⇐⇒ ∃n∃z ∈ 2ω (x = s2n+1,0

_(0)_z&y = s2n+1,1

_(1)_z).

Miller

_(1)_z),Hodd(x, y) ⇐⇒ ∃n∃z ∈ 2ω (x = s2n+1,0

_(0)_z&y = s2n+1,1

_(1)_z).

Miller

Local G0-dichotomy

If G is a graph and E an equivalence relation, both analytic, thenexactly one of the following hold:

1 There is a smooth equivalence relation F ⊇ E and a Borelmeasurable ω-coloring of F ∩G.

2 (Geven,Hodd) ≤c (G,E).

Here F is smooth means that there is a Borel reduction of F toequality on R.

Miller

Local G0-dichotomy

If G is a graph and E an equivalence relation, both analytic, thenexactly one of the following hold:

1 There is a smooth equivalence relation F ⊇ E and a Borelmeasurable ω-coloring of F ∩G.

Here F is smooth means that there is a Borel reduction of F toequality on R.

Harrington-Kechris-Louveau

As a corollary:

Glimm-Effros dichotomy

If E is a bi-analytic equivalence relation, then exactly one of thefollowing hold:

1 E is smooth. (|R/E| ≤ |R| and this is witnessed by a Borelmap.)

2 (Ec0, E0) ≤c (Ec, E). (|R/E0| ≤ |R/E| and this is witnessedby a continuous map.)

As a corollary:

Glimm-Effros dichotomy

If E is a bi-analytic equivalence relation, then exactly one of thefollowing hold:

1 E is smooth. (|R/E| ≤ |R| and this is witnessed by a Borelmap.)

2 (Ec0, E0) ≤c (Ec, E). (|R/E0| ≤ |R/E| and this is witnessedby a continuous map.)

An infinite dimensional version

G0(ω)-dichotomy (Lecomte, Miller)

For G an ω-dimensional κ-Suslin hypergraph, for any strictlyincreasing z ∈ ωω, exactly one of the following occurs:

1 There is a < κ+-Borel κ-coloring of G.

2 G0(ω) � Xz ≤c G.

Here Xz is the comeager dense Gδ set

Xz = {x | ∃∞n [x � n ∈ z(n)n]}.

Lecomte has also shown that the result fails if we remove therestriction to Xz.

2 G0(ω) � Xz ≤c G.

Xz = {x | ∃∞n [x � n ∈ z(n)n]}.

2 G0(ω) � Xz ≤c G.

Xz = {x | ∃∞n [x � n ∈ z(n)n]}.

∞-Borel sets

To state our results (with Ketchersid), it is best to work in afragment of AD+. We need the notion of ∞-Borel sets.Essentially, in the iterative definition of the Borel hierarchy, therequirement that the unions and intersections being taken becountable is weakened to well-orderable.Since we work without choice, rather than the sets themselves, weare more interested in their actual construction. Define the classbc<κ of < κ-Borel codes, for κ a cardinal, as the collection ofwell-founded trees on γ < κ describing how to build a set of realsby taking well-ordered unions and complements from basic sets.

∞-Borel sets

To state our results (with Ketchersid), it is best to work in afragment of AD+. We need the notion of ∞-Borel sets.Essentially, in the iterative definition of the Borel hierarchy, therequirement that the unions and intersections being taken becountable is weakened to well-orderable.Since we work without choice, rather than the sets themselves, weare more interested in their actual construction. Define the classbc<κ of < κ-Borel codes, for κ a cardinal, as the collection ofwell-founded trees on γ < κ describing how to build a set of realsby taking well-ordered unions and complements from basic sets.

∞-Borel sets

More precisely, think of reals as subsets of ω. A code S can beseen as a formula φS in the propositional language L∞,0, where weallow the use of countably many propositional variables pi.The code S describes the set of reals {x :|= φS(x)}, where thesemantics are defined in the standard way, after setting |= pi(x) iffi ∈ x.A < κ-code is then a tree, and can be identified with a set ofordinals bounded below κ. An ∞-Borel code is a < κ-Borel codefor some κ.

∞-Borel sets

Given a < κ-Borel code S, write S(x) to mean “x is in the setcoded by S.” This is very absolute:

S(x) ⇐⇒ Lo(S,x)[S, x] |= φ(S, x),

where o(S, x) = ωCK1 (S, x) is the first admissible over S, x and φis an appropriate Σ1-formula.A < κ-Borel set is the interpretation of a < κ-Borel code. Denoteby B<κ the class of < κ-Borel sets. An ∞-Borel set is a < κ-Borelset for some κ.If Sκ is the set of κ-Suslin sets, then Sκ ⊆ B<κ++ .

∞-Borel sets

S(x) ⇐⇒ Lo(S,x)[S, x] |= φ(S, x),

∞-Borel sets

S(x) ⇐⇒ Lo(S,x)[S, x] |= φ(S, x),

The basic theory

Suslin sets have strong absoluteness properties. In an attempt togeneralize regularity results that hold in the Solovay model orunder AD+ about κ-Suslin sets, we weaken the assumption ofbeing Suslin to simply carrying an ∞-Borel code, and use Los’slemma on ultrapowers to replace the use of absoluteness. For this,it is convenient to work in the following theory:

Definition (BT)

ZF + DCR.

There is a fine σ-complete measure on Pω1(R).

BT holds, for example, in the Solovay model after Levy collapsinga measurable cardinal to ω1 and in models of Turing-determinacy,assuming DCR.

The basic theory

Definition (BT)

ZF + DCR.

The basic theory

Definition (BT)

ZF + DCR.

Regularity properties of ∞-Borel sets

One of the key points about the Solovay model is that every set ofreals in V (R∗) is ∞-Borel, i.e., for all A ∈ P(R)V (R∗), there is aset of ordinals S such that

A(x) ⇐⇒ Lα[S, x] |= φ(S, x)

for appropriate α, φ.In a model of BT, ∞-Borel sets are Lebesgue measurable, have theproperty of Baire, are either countable or contain a perfect set, arecompletely Ramsey, etc.The proofs are exactly as in the Solovay model, because BT impliesthat ω1 is a limit of inaccessibles in any inner model of choice.

A(x) ⇐⇒ Lα[S, x] |= φ(S, x)

For example, if S is a code and P is Cohen forcing, then either forall L[S]-generics x ∈ V , L[S, x] |= S(x), in which case S iscomeager, or else, there is p ∈ P such that for all x ⊇ p,L[S, x] |= ¬S(x), so ¬S is comeager in [p] = {x | p ⊆ x}.Many other regularity properties have an associated forcing notion,for example random real forcing for Lebesgue measure, or Mathiasforcing for completely Ramsey.

Our results apply to models of the strengthening of the axiom ofdeterminacy AD due to Woodin and known as AD+:

Definition (AD+)

< Θ-ordinal determinacy.

All sets of reals are ∞-Borel.

Our results apply to models of the strengthening of the axiom ofdeterminacy AD due to Woodin and known as AD+:

Definition (AD+)

< Θ-ordinal determinacy.

All sets of reals are ∞-Borel.

As motivation for this theory, we have the following result:

Theorem (Woodin)

If M is a transitive model of ZF + AD such that every set of realsin M is Suslin in some transitive model N of ZF + AD with thesame reals, then M |= AD+.

It is still open whether there can be a model of AD which does notalso satisfy AD+.AD+ being in essence an assertion about sets of reals, if AD+

holds in a model M , then it holds in L(P(R))M . We say that anatural model of AD+ is one satisfying V = L(P(R)). It is onthese models that we concentrate.

Theorem (Woodin)

G0-dichotomy

Assuming BT:

G0(d)-dichotomy (d < ω) for ∞-Borel graphs (C-Ketchersid)

Let µ be a fine σ-complete measure on Pω1(R). Suppose G is a< κ-Borel hypergraph with code S. Then exactly one of thefollowing holds:

1 There is a B<κ∞S -measurable κ∞S -coloring.

2 G0(d) ≤c G.

Here κ∞S =∏σ κ

σS/µ where κσS is the first inaccessible of

HODL(S,σ)S . Note that κσS is countable in V .

G0-dichotomy

Assuming BT:

2 G0(d) ≤c G.

G0-dichotomy

Assuming BT:

2 G0(d) ≤c G.

Local G0-dichotomy

Assuming BT:

Local G0-dichotomy (C-Ketchersid)

Let µ be a fine σ-complete measure on Pω1(R). Let E be a< κ-Borel equivalence relation and G be a < κ-Borel graph withcodes SE and SG, respectively. Letting S = SE ⊕ SG, then exactlyone of the following holds:

1 There is B<κ∞S equivalence relation F ⊇ E that isκ∞S -smooth, and a B<κ∞S -measurable, κ∞S -coloring of F ∩ E.

Local G0-dichotomy

Assuming BT:

Local G0-dichotomy (C-Ketchersid)

Let µ be a fine σ-complete measure on Pω1(R). Let E be a< κ-Borel equivalence relation and G be a < κ-Borel graph withcodes SE and SG, respectively. Letting S = SE ⊕ SG, then exactlyone of the following holds:

1 There is B<κ∞S equivalence relation F ⊇ E that isκ∞S -smooth, and a B<κ∞S -measurable, κ∞S -coloring of F ∩ E.

Assuming BT:

Glimm-Effros (C-Ketchersid)

Let µ be a fine σ-complete measure on Pω1(R). If E is a < κ-Borelequivalence relation, then exactly one of the following holds:

1 E is < κ∞S -Borel smooth, that is, |R/E| ≤ |2κ∞S | and this iswitnessed by a B<κ∞S -measurable map.

2 E0 ≤c E. (So |R/E0| ≤ |R/E|.)

Assuming BT:

Glimm-Effros (C-Ketchersid)

Let µ be a fine σ-complete measure on Pω1(R). If E is a < κ-Borelequivalence relation, then exactly one of the following holds:

1 E is < κ∞S -Borel smooth, that is, |R/E| ≤ |2κ∞S | and this iswitnessed by a B<κ∞S -measurable map.

2 E0 ≤c E. (So |R/E0| ≤ |R/E|.)

Cardinal structure

These results allow us to show:

Theorem (C-Ketchersid)

In models of BT of the form V = L(S,R), or in natural models ofAD+, for every set X, exactly one of the following holds:

1 X is well-orderable.

2 X is linearly orderable, but not well-orderable. In this case,|R| = |2ω| ≤ X ≤ |2κ| for some (well-ordered) κ.

3 |R/E0| ≤ |X|.

This extends results of Woodin and Hjorth.

Cardinal structure

3 |R/E0| ≤ |X|.

Cardinal structure

3 |R/E0| ≤ |X|.

Cardinal structure

Recall that E0 is Vitali’s equivalence relation on 2ω:

xE0y iff ∃n ∀m ≥ n (x(m) = y(m)).

One can show that R � 2ω/E0. On the other hand, (Sierpinski)determinacy implies that 2ω/E0 is not linearly orderable. This is asomewhat amusing situation, that a quotient is actually larger thanthe set it comes from.

AD+ implies that |2ω/E0| is an immediate successor of |R|.

Perhaps surprisingly, this was previously known under ADR, butnot in general (not even in L(R)).

Cardinal structure

xE0y iff ∃n ∀m ≥ n (x(m) = y(m)).

Cardinal structure

xE0y iff ∃n ∀m ≥ n (x(m) = y(m)).

Cardinal structure

xE0y iff ∃n ∀m ≥ n (x(m) = y(m)).

G0(ω)

Assuming BT:

Let µ be a fine σ-complete measure on Pω1(R). For G anω-dimensional < κ-Borel hypergraph, for any strictly increasingz ∈ ωω, exactly one of the following holds:

1 There is a < κ∞S -Borel κ∞S -coloring of G.

2 G0(ω) � Xz ≤c G.

G0(ω)

Assuming BT:

Let µ be a fine σ-complete measure on Pω1(R). For G anω-dimensional < κ-Borel hypergraph, for any strictly increasingz ∈ ωω, exactly one of the following holds:

1 There is a < κ∞S -Borel κ∞S -coloring of G.

2 G0(ω) � Xz ≤c G.

The end.

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