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

Abstract

We show under $\mathsf{ZF}+\mathsf{DC}+{\mathsf{AD}}_{\mathbb{R}}$ 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 $\mathsf{ZF}+\mathsf{DC}+{\mathsf{AD}}^{+}$ if we additionally assume that the set of Borel codes for $I$-positive sets is ${\mathbf{\Delta }}_1^2$. If we do not assume $\mathsf{DC}$, the notion of properness becomes obscure as pointed out by Asperó and Karagila. Using the notion of strong properness similar to the one introduced by Bagaria and Bosch, we show under $\mathsf{ZF}+{\mathsf{DC}}_{\mathbb{R}}$ without using $\mathsf{DC}$ 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.