Determinacy and regularity properties for idealized forcings

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Abstract

We show under (Formula presented.) that every set of reals is I-regular for any σ-ideal I on the Baire space (Formula presented.) such that (Formula presented.) is proper. This answers the question of Khomskii [7, Question 2.6.5]. We also show that the same conclusion holds under (Formula presented.) if we additionally assume that the set of Borel codes for I-positive sets is (Formula presented.). If we do not assume (Formula presented.), 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 (Formula presented.) without using (Formula presented.) that every set of reals is I-regular for any σ-ideal I on the Baire space (Formula presented.) such that (Formula presented.) 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.

Original languageEnglish
JournalMathematical Logic Quarterly
DOIs
Publication statusAccepted/In press - 2022

ASJC Scopus subject areas

  • Logic

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