TY - JOUR

T1 - Determinacy and regularity properties for idealized forcings

AU - Ikegami, Daisuke

N1 - Publisher Copyright:
© 2022 Wiley-VCH GmbH.

PY - 2022/8

Y1 - 2022/8

N2 - 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.

AB - 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.

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U2 - 10.1002/malq.202100045

DO - 10.1002/malq.202100045

M3 - Article

AN - SCOPUS:85128856912

SN - 0942-5616

VL - 68

SP - 310

EP - 317

JO - Mathematical Logic Quarterly

JF - Mathematical Logic Quarterly

IS - 3

ER -