TY - JOUR

T1 - Determinacy and regularity properties for idealized forcings

AU - Ikegami, Daisuke

N1 - Funding Information:
The author would like to thank the anonymous referee for careful reading and helpful comments on the paper. He also thanks the Japan Society for the Promotion of Science (JSPS) for its generous support through the grant with JSPS KAKENHI Grant Number 19K03604. He is also grateful to the Sumitomo Foundation for its generous support through Grant for Basic Science Research.
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

VL - 68

SP - 310

EP - 317

JO - Mathematical Logic Quarterly

JF - Mathematical Logic Quarterly

SN - 0942-5616

IS - 3

ER -