Cardinality of wellordered disjoint unions of quotients of smooth equivalence relations

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Abstract

Assume ZF+AD++V=L(P(R)). Let ≈ denote the relation of being in bijection. Let κON and Eα:α<κ be such that for all α<κ, Eα is an equivalence relation on R with all classes countable and R/EαR. Then the disjoint union α<κR/Eα is in bijection with R×κ and α<κR/Eα has the Jónsson property.

Assume ZF+AD++V=L(P(R)). A set X[ω1]<ω1 has a sequence Eα:α<ω1 of equivalence relations on R such that R/EαR and Xα<ω1R/Eα if and only if Rω1 injects into X.

Assume AD. Suppose R[ω1]ω×R is a relation such that for all f[ω1]ω, Rf={xR:R(f,x)} is nonempty and countable. Then there is an uncountable Xω1 and function Φ:[X]ωR which uniformizes R on [X]ω: that is, for all f[X]ω, R(f,Φ(f)).

Under AD, if κ is an ordinal and Eα:α<κ is a sequence of equivalence relations on R with all classes countable, then [ω1]ω does not inject into α<κR/Eα.

Introduction

The original motivation for this work comes from the study of a simple combinatorial property of sets using only definable methods. Let X be a set. For each nω, let [X]=n={fnX:(i,jn)(ijf(i)f(j))}. Let [X]=<ω=nω[X]=n. A set X has the Jónsson property if and only if for every function F:[X]=<ωX, there is some Y such that YX (where ≈ denotes the bijection relation) so that F[[Y]=<ω]X. That is, F can be made to miss an element of X when restricted to the collection of finite injective tuples through some subset Y of X of the same cardinality as X.

Under the axiom of choice, if there is a set with the Jónsson property, then large cardinal principles such as 0 hold. Using a measurable cardinal, one can construct models of ZFC in which 20 is Jónsson and is not Jónsson. Hence assuming the consistency of some large cardinals, the Jónsson property of 20 is independent of ZFC. Using AC, the sets R, Rω1, R×ω1, and R/E0 are all in bijection. (E0 is the equivalence relation defined on R=ω2 by xE0y if and only if (m)(nm)(x(n)=y(n)).)

From a definability perspective, the sets R, R×ω1, Rω1, and R/E0 do not have definable bijections without invoking definable wellorderings of the reals which can exist in canonical inner models like L but in general can not exist if the universe satisfies more regularity properties for sets of reals. For example, there are no injections of R/E0 into R that is induced by a Δ11 reduction Φ:RR of the = relation into the E0 equivalence relation. Such results for the low projective pointclasses can be extended to all sets assuming the axiom of determinacy, AD. Methods that hold in a determinacy setting are often interpreted to be definable methods. Moreover, the extension of AD called AD+ captures this definability setting even better since AD+ implies all sets of reals have a very absolute definition known as the ∞-Borel code.

Kleinberg [14] showed that n has the Jónsson property for all nω under AD. [10] showed that under ZF+AD+V=L(R), every cardinal below Θ has the Jónsson property. (Woodin showed that AD+ alone can prove this result.)

Holshouser and Jackson began the study of the Jónsson property for nonwellorderable sets under AD such as R. In [9], they showed that R and Rω1 have the Jónsson property. They also showed, using the fact that all κ<Θ have the Jónsson property, that R×κ is Jónsson. [5] showed that R/E0 does not have the Jónsson property.

Holshouser and Jackson then asked if the Jónsson property of sets is preserved under the disjoint union operation: If κON and Xα:α<κ is a sequence of sets with the Jónsson property, then does the disjoint union α<κXα have the Jónsson property? (Here ⨆ will always refer to a formal disjoint union in contrast to the ordinary union ⋃.) More specifically, does a disjoint union of sets, each in bijection with R, have the Jónsson property? The determinacy axioms are particularly helpful for studying sets which are surjective images of R. Hence, a natural question would be if Eα:α<κ is a sequence of equivalence relations on R such that for each α, R/Eα is in bijection with R, then does α<κR/Eα have the Jónsson property? An equivalence relation E on R is called smooth if and only if R/E is in bijection with R. (Note that this term is used differently than the ordinary Borel theory which would define E to be smooth if R/E injects into R. This will be referred as being weakly smooth.)

Δ11 equivalence relations with all classes countable are very important objects of study in classical invariant descriptive set theory. One key property that make this investigation quite robust is the Lusin-Novikov countable section uniformization which can be used to show the Feldman-Moore theorem. [2, Theorem 4.13] showed that if Eα:α<κ is a sequence of equivalence relations with all classes countable (not necessarily smooth) and F:[α<κR/Eα]=<ωα<κR/Eα, then there is a perfect tree p on 2 so thatF[[α<κ[p]/Eα]=<ω]α<κR/Eα. (Here R refers to the Cantor space 2ω.) This “pseudo-Jónsson property” would imply the true Jónsson property if α<κ[p]/Eα is in bijection with α<κR/Eα. In general for nonsmooth equivalence relations, this can not be true since, for example, E0 is an equivalence relation with all classes countable and R/E0 is not Jónsson ([5]). When each Eα is the identity relation =, then one can demonstrate these two sets are in bijection. Using this, [2, Theorem 4.15] shows that R×κ has the Jónsson property, where κ is any ordinal, using only classical descriptive set theoretic methods and does not rely on any combinatorial properties of the ordinal κ.

[2, Question 4.14] asked if Eα:α<κ consists entirely of smooth equivalence relations on R with all classes countable and p is any perfect tree on 2, then is α<κR/Eα and α<κ[p]/Eα in bijection? Do such disjoint unions have the Jónsson property? The most natural attempt to show that wellordered disjoint unions of quotients of smooth equivalence relations with all classes countable is Jónsson would be to show it is, in fact, in bijection with R×κ, which has already been shown to possess the Jónsson property.

The computation of the cardinality of wellordered disjoint unions of quotients of smooth equivalence relations on R with all classes countable is the main result of the paper. Under AD+, any equivalence relation on R has an ∞-Borel code. However, for the purpose of this paper, given a sequence of equivalence relations Eα:α<κ on R, one will need to uniformly obtain ∞-Borel codes for each Eα. It is unclear if this is possible under AD+ alone. To obtain this uniformity of ∞-Borel codes, one will need to work with natural models of AD+, i.e. models satisfying ZF+AD++V=L(P(R)). Also the assumption that each equivalence relation has all classes countable is very important. Analogous to the role of the Lusin-Novikov countable section uniformization in the classical setting, Woodin's countable section uniformization under AD+ will play a crucial role.

There are some things that can be said about α<κR/Eα when Eα:α<κ is a sequence of smooth equivalence relations (with possibly uncountable classes). It is immediate that α<κR/Eα will contain a copy of ω1R. Hence ω1R is a lower bound on the cardinality of disjoint unions of quotients of smooth equivalence relations. An example of Holshouser and Jackson (Fact 4.2) produces a sequence Fα:α<ω1 of smooth equivalence relations such that α<ω1R/Fα is in bijection with ω1R. So this lower bound is obtainable. In fact, in natural models of AD+, these two properties are equivalent.

Theorem 6.1 Assume ZF+AD++V=L(P(R)). Suppose X[ω1]<ω1 and Rω1 injects into X. Then there exists a sequence Eα:α<ω1 of smooth equivalence relations on R so that X is in bijection with α<ω1R/Eα. Therefore, X[ω1]<ω1 has a sequence Eα:α<ω1 of smooth equivalence relations such that Xα<ω1R/Eα if and only if Rω1 injects into X.

R×κ is a disjoint union coming from Eα:α<κ where each Eα is the = relation, which is an equivalence relation with all classes countable. However, the proof of Theorem 6.1 uses equivalence relations with uncountable classes. Intuitively, it seems that Rω1, [ω1]ω, and [ω1]<ω1 should not be obtainable using equivalence relations with countable classes. This motivates the conjecture that if Eα:α<κ is a sequence of smooth equivalence relations with all classes countable then the cardinality of α<κR/Eα is R×κ.

Woodin [16] showed that there is elaborate structure of cardinals below [ω1]<ω1. Theorem 6.1 shows that every cardinal above ω1R and below [ω1]<ω1 is a disjoint union of quotients of smooth equivalence relations. A success in Holshouser and Jackson's original goal of establishing the closure of the Jónsson property under disjoint unions would yield the Jónsson property for many cardinals below [ω1]<ω1. On the other hand, it difficult to see how one could establish the Jónsson property for every set that appears in this rich cardinal structure using solely the manifestation of these sets as a disjoint union of quotients of smooth equivalence relations.

The main tool for computing the cardinality of wellordered disjoint union of quotients of smooth equivalence relations with all countable classes is the Woodin's perfect set dichotomy which generalizes the Silver's dichotomy for Π11 equivalence relations. This perfect set dichotomy states that under AD+, for any equivalence relation E on R, either (i) R/E is wellorderable or (2) R injects into R/E. Section 3 is dedicated to proving this result. A detailed analysis of the proof of this result will be needed for this paper. The proof for case (i) yields a uniform procedure which takes an ∞-Borel code for E and gives a wellordering of R/E. Moreover, it shows that in this case, R/EODS, where S is the ∞-Borel code for E. Under ZF+AD++V=L(P(R)), this will give a more general countable section uniformization (Fact 3.4). It will be seen that in the proof of case (ii), the injection of R will depend on certain parameters. If these parameters could be found uniformly for each equivalence relation from the sequence Eα:α<κ, then the proof in case (ii) can uniformly produce injections of R into each R/Eα. Together, one would get an injection of R×κ into α<κR/Eα. In general this can not be done; for instance using the example from Fact 4.2. However, this can be done when all the equivalence relations are smooth and have all classes countable. This shows the following:

Theorem 4.5 Assume ZF+AD++V=L(P(R)). Let κON and Eα:α<κ be a sequence of smooth equivalence relations on R with all classes countable. Then R×κ injects into α<κR/Eα.

This shows that R×κ is a lower bound for the cardinality of α<κR/Eα.

Section 5 will provide the proof of the relevant half of Hjorth's generalized E0-dichotomy. Again, what is important from this result is the observation that if R/E0 does not inject into R/E, then there is a wellordered separating family for E defined uniformly from the ∞-Borel code for E. If ZF+AD++V=L(P(R)) holds, then one has a uniform sequence of ∞-Borel codes for the sequence of equivalence relations Eα:α<κ, where each Eα is smooth. Using the argument of Hjorth's dichotomy, one obtains uniformly a separating family for each Eα. This gives a sequence of injections of each R/Eα into P(δ) where δ is a possibly very large ordinal. If Eα:α<κ consists entirely of equivalence relations with all classes countable, then the generalized countable section uniformization can be used to uniformly obtain a selector for each R/Eα. This gives the desired injection into R×κ.

Theorem 5.4 Assume ZF+AD++V=L(P(R)). Let κ be an ordinal and Fα:α<κ be a sequence of smooth equivalence relations on R with all classes countable. Then there is an injection of α<κR/Fα into R×κ.

Theorem 5.8 Assume ZF+AD++V=L(P(R)). Let κON and Eα:α<κ be a sequence of smooth equivalence relations on R with all classes countable. Then α<κR/EαR×κ and hence α<κR/Eα has the Jónsson property.

[ω1]ω is the collection of increasing functions f:ωω1. [2] proved under AD+ that [ω1]ω does not inject into α<κR/Eα when Eα:α<κ is a sequence of equivalence relations on R with all classes countable. (Of course, Theorem 5.8 asserts that such a disjoint union is in bijection with R×κ under the assumption ZF+AD++V=L(P(R)).) The key ingredient is the ability to uniformize relations R[ω1]ω×R such that for all f[ω1]ω, Rf={xR:R(f,x)} is nonempty and countable. Such a full uniformization is provable under AD+. A careful inspection of the argument will show that one only needs to uniformize this relation on some Z[ω1]ω such that Z[ω1]ω to show that no such injection exists. [2, Question 4.21] asks whether such an almost full uniformization is provable in AD.

It should be noted that if one drops the demand that Rf be countable, then one cannot prove this in general. (See the discussion in Section 7.) The final section will show that such an almost full countable section uniformization for relations on [ω1]ω×R is provable in AD.

Theorem 7.4 (ZF+AD) Let R[ω1]ω×R be such that for all f[ω1]ω, Rf is nonempty and countable. Then there exists some uncountable Xω1 and function Ψ which uniformizes R on [X]ω: For f[X]ω, R(f,Ψ(f)).

Corollary 7.5 (ZF+AD) Let Eα:α<κ be a sequence of equivalence relations on R with all classes countable, then [ω1]ω does not inject into α<κR/Eα.

These methods also show that for an arbitrary function Φ:[ω1]ωR, one can find some uncountable Xω1 and some reals σ and w so that Φ(f)L[σ,w,f], for all f[X]ω.

Theorem 7.6 (ZF+AD). Let Φ:[ω1]ωR be a function. Then there is an uncountable Xω1, reals σ,wR, and a formula ϕ so that for all f[X]ω, Φ(f)L[σ,w,f] and for all zR, z=Φ(f) if and only if L[σ,w,f,z]ϕ(σ,w,f,z).

Section snippets

Basics

See [3, Section 7 and 8] for more information on AD+, ∞-Borel codes, and Vopěnka forcing.

Definition 2.1

Let S be a set of ordinals and φ be a formula in the language of set theory. (S,φ) is called an ∞-Borel code. For each nω, B(S,φ)n={xRn:L[S,x]φ(S,x)} is the subset of Rn coded by (S,φ).

A set ARn is ∞-Borel if and only if A=B(S,φ)n for some ∞-Borel code (S,φ). (S,φ) is called an ∞-Borel code for A.

Definition 2.2

[17, Section 9.1] AD+ consists of the following statements:

(1) DCR.

(2) Every AR has an ∞-Borel code.

(3) For

Perfect set dichotomy and wellorderable section uniformization

The Silver's dichotomy ([15] and [7]) states the every Π11 equivalence relation E on R has countably many equivalence classes (R/E is hence wellorderable) or E has a perfect set of pairwise E-inequivalent elements (R injects into R/E). Woodin's perfect set dichotomy states: under AD+, for every equivalence relation E on R, either R/E is wellorderable or R injects into R/E. As a consequence, every set which is a surjective image of R, either the set contains a copy of R or is wellorderable. (In

Lower bound on cardinality

The following section gives a lower bound on the cardinality of disjoint unions of smooth equivalence relations on R. (Later it will be shown that this fact characterizes those subsets of [ω1]<ω1 which are in bijection with a disjoint union of quotients of smooth equivalence relations.)

Fact 4.1

(ZF). Let κ>0 be an ordinal. Let Gα:α<κ be a sequence of smooth equivalence relations. Then Rκ injects into α<κR/Gα.

Proof

Let 0¯R denote the constant 0 function. Let Φ:RR/G0 be an injection so that the image of Φ

Upper bound on cardinality

Definition 5.1

Let E be an equivalence relation on R. Let S be a collection of nonempty subsets of R. S is a separating family for E if and only if for all x,yR, xEy if and only if for all AS, xAyA. In other words, for all x,yR, ¬(xEy) if and only if there is an AS which separate x and y in the sense that either (xAyA) or (xAyA).

Definition 5.2

E0 is the equivalence relation on 2ω defined by xE0y if and only if (m)(nm)(x(n)=y(n)).

The following is Hjorth's E0-dichotomy in AD+ which generalizes the classical E0

Disjoint union of quotients of smooth equivalence with uncountable classes

This section will show that in natural models of AD+ many subsets below [ω1]<ω1 are in bijection with disjoint unions of quotients of smooth equivalence relations on R. It will be shown that any subset of [ω1]<ω1 that contains Rω1 can be written in this way. The argument is similar to the example Fα:α<ω1 produced in the proof of Fact 4.2. Note that each Fα has one uncountable equivalence class that holds the reals that are not used for coding. In the following argument, the existence of a

Almost full countable section uniformization for [ω1]ω×R

This section will show that |[ω1]ω| is not below |α<κR/Eα| if Eα:α<κ is a sequence of equivalence relations on R with all classes countable under just AD. Note that by Theorem 5.8, ZF+AD++V=L(P(R)) is capable of proving that such a disjoint union is in bijection with R×κ. It is much more evident that [ω1]ω does not inject into R×κ.

Fact 7.1

([2]) (ZF+AD) Let κ be an ordinal. Let Eα:α<κ be a sequence of equivalence relations on R. Let Φ:[ω1]ωα<κR/Eα. Let R[ω1]ω×R be defined by R(f,x)xΦ(f). If

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The first author was supported by NSF grant DMS-1703708. The second author was supported by NSF grant DMS-1800323.

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