Determinacy and regularity properties for idealized forcings
Determinacy and regularity properties for idealized forcings
复制标题
理想化力的确定性和规律性特性
DOI：
10.1002/malq.202100045
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发表时间：
2022
影响因子：
0.3
通讯作者：
Ikegami Daisuke
中科院分区：
文献类型：
--
作者：
Kubota Sho;Segawa Etsuo;Ikegami Daisuke
We show under ZF+DC+ADR$\sf {ZF}+ \sf {DC}+ \sf {AD}_\mathbb {R}$ that every set of reals isI‐regular for any σ‐idealIon the Baire space ωω$\omega ^{\omega }$ such that PI$\mathbb {P}_I$ is proper. This answers the question of Khomskii [7, Question 2.6.5]. We also show that the same conclusion holds under ZF+DC+AD+$\sf {ZF}+ \sf {DC}+ \sf {AD}^+$ if we additionally assume that the set of Borel codes forI‐positive sets is Δ˜12$\undertilde{\mathbf {\Delta }}^2_1$. If we do not assume DC$\sf {DC}$, 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+DCR$\sf {ZF}+ \sf {DC}_{\mathbb {R}}$ without using DC$\sf {DC}$ that every set of reals isI‐regular for any σ‐idealIon the Baire space ωω$\omega ^{\omega }$ such that PI$\mathbb {P}_I$ 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.
We show under ZF+DC+ADR$\sf {ZF}+ \sf {DC}+ \sf {AD}_\mathbb {R}$ that every set of reals isI‐regular for any σ‐idealIon the Baire space ωω$\omega ^{\omega }$ such that PI$\mathbb {P}_I$ is proper. This answers the question of Khomskii [7, Question 2.6.5]. We also show that the same conclusion holds under ZF+DC+AD+$\sf {ZF}+ \sf {DC}+ \sf {AD}^+$ if we additionally assume that the set of Borel codes forI‐positive sets is Δ˜12$\undertilde{\mathbf {\Delta }}^2_1$. If we do not assume DC$\sf {DC}$, 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+DCR$\sf {ZF}+ \sf {DC}_{\mathbb {R}}$ without using DC$\sf {DC}$ that every set of reals isI‐regular for any σ‐idealIon the Baire space ωω$\omega ^{\omega }$ such that PI$\mathbb {P}_I$ 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.
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影响因子：
0.8
作者：
D. Ikegami
通讯作者：
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DOI：
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发表时间：
2007
期刊：
影响因子：
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Y. D. Khomskii
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Y. D. Khomskii
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影响因子：
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作者：
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通讯作者：
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影响因子：
1.1
作者：
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DOI：
10.1017/s0025557200177484
发表时间：
2005
期刊：
The Mathematical Gazette
影响因子：
--
作者：
G. Leversha
通讯作者：
G. Leversha