Determinacy and regularity properties for idealized forcings

Abstract

We show under $\mathrm{ZF}+\mathrm{DC}+{\mathrm{AD}}_R$ that every set of reals is I-regular for any σ-ideal I on the Baire space ${\omega }^{\omega }$ such that $P_I$ is proper. This answers the question of Khomskii [7, Question 2.6.5]. We also show that the same conclusion holds under $\mathrm{ZF}+\mathrm{DC}+{\mathrm{AD}}^{+}$ if we additionally assume that the set of Borel codes for I-positive sets is ${\underset{˜}{\Delta }}_1^2$. If we do not assume $\mathrm{DC}$, the notion of properness becomes obscure as pointed out by Asperó and Karagila [1]. Using the notion of strong properness similar to the one introduced by Bagaria and Bosch [2], we show under $\mathrm{ZF}+{\mathrm{DC}}_R$ without using $\mathrm{DC}$ that every set of reals is I-regular for any σ-ideal I on the Baire space ${\omega }^{\omega }$ such that $P_I$ is strongly proper assuming every set of reals is ∞-Borel and there is no ω₁-sequence of distinct reals. In particular, the same conclusion holds in a Solovay model.