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

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.