The countable admissible ordinal equivalence relation

Abstract

Let F_(ω1) be the countable admissible ordinal equivalence relation defined on ^ω2 by x F_(ω1) y if and only if ω_1^x=ω_1^y. Some invariant descriptive set theoretic properties of F_(ω1) will be explored using infinitary logic in countable admissible fragments as the main tool. Marker showed F_(ω1) is not the orbit equivalence relation of a continuous action of a Polish group on ^ω2. Becker stengthened this to show F_(ω1) is not even the orbit equivalence relation of a Δ_1^1 action of a Polish group. However, Montalbán has shown that F_(ω1) is Δ_1^1 reducible to an orbit equivalence relation of a Polish group action, in fact, F_(ω1) is classifiable by countable structures. It will be shown here that F_(ω1) must be classified by structures of high Scott rank. Let E_(ω1) denote the equivalence of order types of reals coding well-orderings. If E and F are two equivalence relations on Polish spaces X and Y, respectively, E ≤ aΔ_1^1 F denotes the existence of a Δ_1^1 function f:X→Y which is a reduction of E to F, except possibly on countably many classes of E. Using a result of Zapletal, the existence of a measurable cardinal implies E_(ω1) ≤ aΔ_1^1 F_(ω1). However, it will be shown that in Gödel's constructible universe L (and set generic extensions of L), E_(ω1) ≤ aΔ_1^1 F_(ω1) is false. Lastly, the techniques of the previous result will be used to show that in L (and set generic extensions of L), the isomorphism relation induced by a counterexample to Vaught's conjecture cannot be Δ_1^1 reducible to F_(ω1). This shows the consistency of a negative answer to a question of Sy-David Friedman.

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