Cardinality of wellordered disjoint unions of quotients of smooth equivalence relations☆
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 , let . Let . A set X has the Jónsson property if and only if for every function , there is some Y such that (where ≈ denotes the bijection relation) so that . 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 hold. Using a measurable cardinal, one can construct models of in which is Jónsson and is not Jónsson. Hence assuming the consistency of some large cardinals, the Jónsson property of is independent of . Using , the sets , , , and are all in bijection. ( is the equivalence relation defined on by if and only if .)
From a definability perspective, the sets , , , and 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 into that is induced by a reduction of the = relation into the equivalence relation. Such results for the low projective pointclasses can be extended to all sets assuming the axiom of determinacy, . Methods that hold in a determinacy setting are often interpreted to be definable methods. Moreover, the extension of called captures this definability setting even better since implies all sets of reals have a very absolute definition known as the ∞-Borel code.
Kleinberg [14] showed that has the Jónsson property for all under . [10] showed that under , every cardinal below Θ has the Jónsson property. (Woodin showed that alone can prove this result.)
Holshouser and Jackson began the study of the Jónsson property for nonwellorderable sets under such as . In [9], they showed that and have the Jónsson property. They also showed, using the fact that all have the Jónsson property, that is Jónsson. [5] showed that 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 and is a sequence of sets with the Jónsson property, then does the disjoint union 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 , have the Jónsson property? The determinacy axioms are particularly helpful for studying sets which are surjective images of . Hence, a natural question would be if is a sequence of equivalence relations on such that for each α, is in bijection with , then does have the Jónsson property? An equivalence relation E on is called smooth if and only if is in bijection with . (Note that this term is used differently than the ordinary Borel theory which would define E to be smooth if injects into . This will be referred as being weakly smooth.)
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 is a sequence of equivalence relations with all classes countable (not necessarily smooth) and , then there is a perfect tree p on 2 so that (Here refers to the Cantor space .) This “pseudo-Jónsson property” would imply the true Jónsson property if is in bijection with . In general for nonsmooth equivalence relations, this can not be true since, for example, is an equivalence relation with all classes countable and is not Jónsson ([5]). When each is the identity relation =, then one can demonstrate these two sets are in bijection. Using this, [2, Theorem 4.15] shows that 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 consists entirely of smooth equivalence relations on with all classes countable and p is any perfect tree on 2, then is and 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 , 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 with all classes countable is the main result of the paper. Under , any equivalence relation on has an ∞-Borel code. However, for the purpose of this paper, given a sequence of equivalence relations on , one will need to uniformly obtain ∞-Borel codes for each . It is unclear if this is possible under alone. To obtain this uniformity of ∞-Borel codes, one will need to work with natural models of , i.e. models satisfying . 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 will play a crucial role.
There are some things that can be said about when is a sequence of smooth equivalence relations (with possibly uncountable classes). It is immediate that will contain a copy of . Hence 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 of smooth equivalence relations such that is in bijection with . So this lower bound is obtainable. In fact, in natural models of , these two properties are equivalent.
Theorem 6.1 Assume . Suppose and injects into X. Then there exists a sequence of smooth equivalence relations on so that X is in bijection with . Therefore, has a sequence of smooth equivalence relations such that if and only if injects into X.
is a disjoint union coming from where each 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 , , and should not be obtainable using equivalence relations with countable classes. This motivates the conjecture that if is a sequence of smooth equivalence relations with all classes countable then the cardinality of is .
Woodin [16] showed that there is elaborate structure of cardinals below . Theorem 6.1 shows that every cardinal above and below 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 . 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 equivalence relations. This perfect set dichotomy states that under , for any equivalence relation E on , either (i) is wellorderable or (2) injects into . 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 . Moreover, it shows that in this case, , where S is the ∞-Borel code for E. Under , 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 will depend on certain parameters. If these parameters could be found uniformly for each equivalence relation from the sequence , then the proof in case (ii) can uniformly produce injections of into each . Together, one would get an injection of into . 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 . Let and be a sequence of smooth equivalence relations on with all classes countable. Then injects into .
This shows that is a lower bound for the cardinality of .
Section 5 will provide the proof of the relevant half of Hjorth's generalized -dichotomy. Again, what is important from this result is the observation that if does not inject into , then there is a wellordered separating family for E defined uniformly from the ∞-Borel code for E. If holds, then one has a uniform sequence of ∞-Borel codes for the sequence of equivalence relations , where each is smooth. Using the argument of Hjorth's dichotomy, one obtains uniformly a separating family for each . This gives a sequence of injections of each into where δ is a possibly very large ordinal. If 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 . This gives the desired injection into .
Theorem 5.4 Assume . Let κ be an ordinal and be a sequence of smooth equivalence relations on with all classes countable. Then there is an injection of into .
Theorem 5.8 Assume . Let and be a sequence of smooth equivalence relations on with all classes countable. Then and hence has the Jónsson property.
is the collection of increasing functions . [2] proved under that does not inject into when is a sequence of equivalence relations on with all classes countable. (Of course, Theorem 5.8 asserts that such a disjoint union is in bijection with under the assumption .) The key ingredient is the ability to uniformize relations such that for all , is nonempty and countable. Such a full uniformization is provable under . A careful inspection of the argument will show that one only needs to uniformize this relation on some such that to show that no such injection exists. [2, Question 4.21] asks whether such an almost full uniformization is provable in .
It should be noted that if one drops the demand that 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 is provable in .
Theorem 7.4 Let be such that for all , is nonempty and countable. Then there exists some uncountable and function Ψ which uniformizes R on : For , .
Corollary 7.5 Let be a sequence of equivalence relations on with all classes countable, then does not inject into .
These methods also show that for an arbitrary function , one can find some uncountable and some reals σ and w so that , for all .
Theorem 7.6 . Let be a function. Then there is an uncountable , reals , and a formula ϕ so that for all , and for all , if and only if .
Section snippets
Basics
See [3, Section 7 and 8] for more information on , ∞-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. is called an ∞-Borel code. For each , is the subset of coded by . A set is ∞-Borel if and only if for some ∞-Borel code . is called an ∞-Borel code for A.
Definition 2.2 [17, Section 9.1] consists of the following statements: (1) . (2) Every has an ∞-Borel code. (3) For
Perfect set dichotomy and wellorderable section uniformization
The Silver's dichotomy ([15] and [7]) states the every equivalence relation E on has countably many equivalence classes ( is hence wellorderable) or E has a perfect set of pairwise E-inequivalent elements ( injects into ). Woodin's perfect set dichotomy states: under , for every equivalence relation E on , either is wellorderable or injects into . As a consequence, every set which is a surjective image of , either the set contains a copy of 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 . (Later it will be shown that this fact characterizes those subsets of which are in bijection with a disjoint union of quotients of smooth equivalence relations.)
Fact 4.1 . Let be an ordinal. Let be a sequence of smooth equivalence relations. Then injects into .
Proof Let denote the constant 0 function. Let be an injection so that the image of Φ
Upper bound on cardinality
Definition 5.1 Let E be an equivalence relation on . Let be a collection of nonempty subsets of . is a separating family for E if and only if for all , if and only if for all , . In other words, for all , if and only if there is an which separate x and y in the sense that either or .
Definition 5.2 is the equivalence relation on defined by if and only if .
The following is Hjorth's -dichotomy in which generalizes the classical
Disjoint union of quotients of smooth equivalence with uncountable classes
This section will show that in natural models of many subsets below are in bijection with disjoint unions of quotients of smooth equivalence relations on . It will be shown that any subset of that contains can be written in this way. The argument is similar to the example produced in the proof of Fact 4.2. Note that each 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
This section will show that is not below if is a sequence of equivalence relations on with all classes countable under just . Note that by Theorem 5.8, is capable of proving that such a disjoint union is in bijection with . It is much more evident that does not inject into .
Fact 7.1 ([2]) Let κ be an ordinal. Let be a sequence of equivalence relations on . Let . Let be defined by . 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.