The Countable Admissible Ordinal Equivalence Relation

2016/01/28 by Chan, William
#FOS: Mathematics #Logic (math.LO)

paper · doi:10.48550/arxiv.1601.07924

Abstract

Let F_ω1 be the countable admissible ordinal equivalence relation defined on ^ω2 by x F_ω1 y if and only if ω₁^x = ω₁^y. It will be shown that F_ω1 is classifiable by countable structures and must be classified by structures of high Scott rank. If E and F are equivalence relations, then E is almost Borel reducible to F if and only if there is a Borel reduction of E to F, except possibly on countably many E-classes. Let E_ω1 denote the equivalence of order types of reals coding well-orderings. It will be shown that in the constructible universe L and set generic extensions of L, E_ω1 is not almost Borel reducible to F_ω1, although a result of Zapletal implies such an almost Borel reduction exists if there is a measurable cardinal. Lastly, it will be shown that the isomorphism relation induced by a counterexample to Vaught's conjecture cannot be Borel reducible to F_ω1 in L and set generic extensions of L. This shows the consistency of a negative answer to a question of Sy-David Friedman.

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