The Effective Theory of Borel Equivalence Relations

TL;DR

This paper explores the effective analogues of classical results in Borel equivalence relations, revealing a complex structure that can be better understood using Kleene's O, and introduces a key lemma involving effectively Borel sets and Barwise compactness.

Contribution

It extends classical Borel equivalence relation results to the effective setting, demonstrating how Kleene's O restores the known structure and establishing a novel lemma about effectively Borel sets.

Findings

Effective Borel equivalence relations form a complex structure.

Kleene's O parameter restores the classical dichotomies.

Existence of two effectively Borel sets with incomparable ranges.

Abstract

The study of Borel equivalence relations under Borel reducibility has developed into an important area of descriptive set theory. The dichotomies of Silver and Harrington-Kechris-Louveau show that with respect to Borel reducibility, any Borel equivalence relation strictly above equality on $\omega$ is above equality on ${\cal P}(\omega)$, the power set of $\omega$, and any Borel equivalence relation strictly above equality on the reals is above equality modulo finite on ${\cal P}(\omega)$. In this article we examine the effective content of these and related results by studying effectively Borel equivalence relations under effectively Borel reducibility. The resulting structure is complex, even for equivalence relations with finitely many equivalence classes. However use of Kleene's $O$ as a parameter is sufficient to restore the picture from the noneffective setting. A key lemma is the existence of two effectively Borel sets of reals, neither of which contains the range of the other under any effectively Borel function; the proof of this result applies Barwise compactness to a deep theorem of Harrington establishing for any recursive ordinal $\alpha$ the existence of $\Pi^0_1$ singletons whose $\alpha$-jumps are Turing incomparable.