$I$-regularity, determinacy, and $\infty$-Borel sets of reals

TL;DR

This paper demonstrates that under certain set-theoretic axioms, all sets of reals are regular with respect to any proper or strongly proper sigma-ideal, extending classical regularity results to broader contexts.

Contribution

It establishes the regularity of all sets of reals for any proper or strongly proper sigma-ideal under various axioms, including $ ext{ZF} + ext{DC} + ext{AD}_ ext{R}$ and $ ext{ZF} + ext{DC}_ ext{R}$, addressing open questions.

Findings

All sets of reals are $I$-regular under $ ext{ZF} + ext{DC} + ext{AD}_ ext{R}$ for proper $\sigma$-ideals.

The same regularity holds under $ ext{ZF} + ext{DC} + ext{AD}^+$ with additional conditions on Borel codes.

In models without $ ext{DC}$, strong properness ensures regularity assuming all sets are $ ext{infty}$-Borel and no $ ext{omega}_1$-sequence of reals exists.

Abstract

We show under $\sf{ZF} + \sf{DC} + \sf{AD}_{\mathbb{R}}$ that every set of reals is $I$-regular for any $\sigma$-ideal $I$ on the Baire space $\omega^{\omega}$ such that $\mathbb{P}_I$ is proper. This answers the question of Khomskii. We also show that the same conclusion holds under $\sf{ZF} + \sf{DC} + \sf{AD}^+$ if we additionally assume that the set of Borel codes for $I$-positive sets is $\mathbf{\Delta}^2_1$. If we do not assume $\sf{DC}$, the notion of properness becomes obscure as pointed out by Asper\'{o} and Karagila. Using the notion of strong properness similar to the one introduced by Bagaria and Bosch, we show under $\sf{ZF} + \sf{DC}_{\mathbb{R}}$ without using $\sf{DC}$ that every set of reals is $I$-regular for any $\sigma$-ideal $I$ on the Baire space $\omega^{\omega}$ such that $\mathbb{P}_I$ is strongly proper assuming every set of reals is $\infty$-Borel and there is no $\omega_1$-sequence of distinct reals. In particular, the same conclusion holds in a Solovay model.