The countable admissible ordinal equivalence relation

Let Fω1 be the countable admissible ordinal equivalence relation defined on 2ω by xFω1y if and only if ω1x=ω1y. Some invariant descriptive set theoretic properties of Fω1 will be explored using infinitary logic in countable admissible fragments as the main tool. Marker showed Fω1 is not the orbit equivalence relation of a continuous action of a Polish group on 2ω. Becker stengthened this to show Fω1 is not even the orbit equivalence relation of a Δ11 action of a Polish group. However, Montalbán has shown that Fω1 is Δ11 reducible to an orbit equivalence relation of a Polish group action, in fact, Fω1 is classifiable by countable structures. It will be shown here that Fω1 must be classified by structures of high Scott rank. Let Eω1 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≤aΔ11F denotes the existence of a Δ11 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 Eω1≤aΔ11Fω1. However, it will be shown that in Gödel's constructible universe L (and set generic extensions of L), Eω1≤aΔ11Fω1 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 Δ11 reducible to Fω1. This shows the consistency of a negative answer to a question of Sy-David Friedman.