Measurability and Perfect Set Theorems for Equivalence Relations with Small Classes

TL;DR

This paper investigates conditions under which certain definable equivalence relations with small classes necessarily have many classes, linking these conditions to measurability and the Baire property in descriptive set theory.

Contribution

It establishes equivalences between the existence of perfect sets of inequivalent elements and measurability properties for $oldsymbol{ riangle^1_2}$ and $oldsymbol{oldsymbol{ abla}^1_2}$ equivalence relations.

Findings

Positive results for $oldsymbol{ riangle^1_2}$ relations depend on $I$-measurability of $oldsymbol{ riangle^1_2}$ sets.

For $oldsymbol{oldsymbol{ abla}^1_2}$ relations, the existence of perfect sets is equivalent to the Baire property.

The paper links set-theoretic properties of equivalence relations to classical regularity properties of definable sets.

Abstract

We ask whether $\mathbf{\Delta^1_2}$ or $\mathbf{\Sigma^1_2}$ equivalence relations with $I$-small classes for $I$ a $\sigma$-ideal must have perfectly many classes. We show that for a wide class of ccc $\sigma$-ideals, a positive answer for $\mathbf{\Delta^1_2}$ equivalence relations is equivalent to the $I$-measurability of $\mathbf{\Delta^1_2}$ sets. However, the analogous statement for $\mathbf{\Sigma^1_2}$ equivalence relations is false: $\mathbf{\Sigma^1_2}$ equivalence relations with meager classes have a perfect set of pairwise inequivalent elements if and only if $\mathbf{\Delta^1_2}$ sets have the Baire property.