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

Abstract

We show under ZF+DC+AD R that every set of reals is I-regular for any σ-ideal I on the Baire space ω ω such that P I is proper. This answers the question of Khomskii [5, Question 2.6.5]. We also show that the same conclusion holds under ZF + DC + AD + if we additionally assume that the set of Borel codes for I-positive sets is ∆ 21 . If we do not assume 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 ZF + DC R without using DC that every set of reals is I-regular for any σ-ideal I on the Baire space ω ω such that P I is strongly proper assuming every set of reals is ∞-Borel and there is no ω 1 -sequence of distinct reals. In particular, the same conclusion holds in a Solovay model.