On the Equality Relation Modulo a Countable Set
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On the Equality Relation Modulo a Countable Set V. G. Kanovei1* and V. A. Lyubetsky1** 1
Institute for Information Transmission Problems of Russian Academy of Sciences (Kharkevich Institute), Moscow, 127051 Russia Received April 15, 2020; in final form, April 15, 2020; accepted May 14, 2020
DOI: 10.1134/S0001434620090357 Keywords: equality modulo a countable set, effective choice.
The equivalence relation Eℵ0 (the equality relation modulo a countable set) is defined as follows: X Eℵ0 Y,
if the symmetric difference XΔY is at most countable.
Here X and Y are sets in the Baire space ω ω , although all that is stated below remains true for the real line and, in general, for any perfect Polish space. The axiom of choice AC allows us to choose a particular element in each Eℵ0 -equivalence class; that is, there exists a function s : P(ω ω ) → P(ω ω ) satisfying s(X) Eℵ0 X for all sets X ⊆ ω ω and s(Y ) = s(X) for all sets X, Y ⊆ ω ω satisfying X Eℵ0 Y . Such a function S is called a selector for the relation Eℵ0 ; see [1, Section 12.D]. However, the application of the axiom of choice does not resolve the question of the existence of an effectively defined selector s, that is, the choice of a concrete well-defined set in every Eℵ0 -class of point sets. The answer to this question depends on which point sets we consider. For instance, every class of Eℵ0 -equivalence of closed sets X ⊆ ω ω contains a unique perfect set, which we can take as s(X), obtaining an effectively defined selector. The following theorem of ours extends this result to the much broader class Δ02 of those sets that are simultaneously Fσ and Gδ . Theorem. There is an effectively defined selector for the relation Eℵ0 on the Δ02 sets in the Baire space. The theorem gives the best possible result, since already for the next (according to the volume of sets) Borel class Fσ , there are generally no effectively definable selectors. This is a consequence of the result, recently obtained in [2, 5.5], that ZFC is not strong enough to define an effectively definable selector for the relation Eℵ0 on the class of all Fσ sets, which is wider than Δ02 .1 The proof of the theorem uses the following Lemma. As usual, X denotes the topological closure of a set X. Lemma. If X is a countable Gδ set in a Polish space, then the closure X of X is countable. Therefore, if the Δ02 sets X and Y satisfy X Eℵ0 Y , then X Eℵ0 Y . Proof. Otherwise, X would be a countable dense Gδ set in the uncountable Polish space X, which is a contradiction. *
E-mail: [email protected] E-mail: [email protected] 1 To be more exact, the result in [2, 5.5] claims that the relation Eℵ0 on the Fσ sets does not admit a Baire measurable selector. However, it is true in the well-known Solovay model [3] of ZFC that all ROD maps are Baire measurable. The class ROD (real-ordinal definable sets) consists of all sets definable by set-theoretic formulas with ordinals and points of ω ω (that is, reals, in the terminology of modern descriptive set theory) as parameters. The class ROD contains, with a margin, all t
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