The Countable Admissible Ordinal Equivalence Relation

TL;DR

This paper investigates the complexity of the countable admissible ordinal equivalence relation, showing its classification properties, and explores the reducibility relations between various equivalence relations in different set-theoretic universes.

Contribution

It establishes the classification complexity of $F_{\omega_1}$, analyzes almost Borel reducibility between equivalence relations, and demonstrates independence results related to Vaught's conjecture.

Findings

$F_{\omega_1}$ is classifiable by structures of high Scott rank.

In $L$, $E_{\omega_1}$ is not almost Borel reducible to $F_{\omega_1}$.

The isomorphism relation of a Vaught's conjecture counterexample cannot be Borel reduced to $F_{\omega_1}$ in $L$.

Abstract

Let $F_{\omega_1}$ be the countable admissible ordinal equivalence relation defined on ${}^\omega 2$ by $x \ F_{\omega_1} \ y$ if and only if $\omega_1^x = \omega_1^y$. It will be shown that $F_{\omega_1}$ 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 $E_{\omega_1}$ 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 $L$, $E_{\omega_1}$ is not almost Borel reducible to $F_{\omega_1}$, 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 $F_{\omega_1}$ in $L$ and set generic extensions of $L$. This shows the consistency of a negative answer to a question of Sy-David Friedman.