$\mathbf{Σ_{3}^{1}}$ Generic Absoluteness, Measurability and Perfect Set Theorems for Equivalence Relations with small classes

Ohad Drucker

Published 2016-05-19Version 1

We introduce a regularity property of equivalence relations - the existence of a perfect set of pairwise inequivalent elements when all equivalence classes are $I$ - small with respect to a $\sigma$ ideal $I$. We show this regularity property for $\mathbf{\Sigma^1_2}$ and $\mathbf{\Delta^1_2}$ equivalence relations is interconnected with $\Sigma^1_3$ generic absoluteness on one hand and transcendence over $L$ on the other hand. For example, we show that given a definable enough provably ccc $\sigma$ - ideal $I$, all $\mathbf{\Delta^1_2}$ equivalence relations with $I$ - small classes have a perfect set of pairwise inequivalent elements if and only if there is $\Sigma^1_3$ $\mathbb{P}_I$ generic absoluteness, if and only if for every real $z$ and $B$ positive there is a $\mathbb{P}_I$ generic real over $L[z]$ in $B$. For the case of the meager ideal, we prove that $\mathbf{\Sigma^1_2}$ equivalence relations with meager classes have a perfect set of pairwise inequivalent elements if and only if for every real $z$ there is a Cohen real over $L[z]$.