Uniformization and Internal Absoluteness

TL;DR

This paper explores the deep connections between uniformization principles, measurability, and absoluteness in the context of forcing notions associated with ideals, establishing new equivalences that extend beyond traditional projective hierarchy levels.

Contribution

It proves that for certain proper ideal forcings, uniformization principles are equivalent to internal absoluteness and measurability combined with 1-step absoluteness, extending known results.

Findings

Uniformization principle is equivalent to internal absoluteness for proper ideal forcings.

Equivalence between measurability and 1-step absoluteness is established.

Results extend the understanding of regularity and absoluteness beyond the second projective level.

Abstract

Measurability with respect to ideals is tightly connected with absoluteness principles for certain forcing notions. We study a uniformization principle that postulates the existence of a uniformizing function on a large set, relative to a given ideal. We prove that for all $\sigma$-ideals $I$ such that the ideal forcing $\mathbb{P}_I$ of Borel sets modulo $I$ is proper, this uniformization principle is equivalent to an absoluteness principle for projective formulas with respect to $\mathbb{P}_I$ that we call internal absoluteness. In addition, we show that it is equivalent to measurability with respect to $I$ together with $1$-step absoluteness for the poset $\mathbb{P}_I$. These equivalences are new even for Cohen and random forcing and they are, to the best of our knowledge, the first precise equivalences between regularity and absoluteness beyond the second level of the projective hierarchy.