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

Daisuke Ikegami

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

Logic 2021-08-20 v2

Authors: Daisuke Ikegami

Abstract

We show under $\sf{ZF} + \sf{DC} + \sf{AD}_{\mathbb{R}}$ that every set of reals is $I$ -regular for any $\sigma$ -ideal $I$ on the Baire space $\omega^{\omega}$ such that $\mathbb{P}_I$ is proper. This answers the question of Khomskii. We also show that the same conclusion holds under $\sf{ZF} + \sf{DC} + \sf{AD}^+$ if we additionally assume that the set of Borel codes for $I$ -positive sets is $\mathbf{\Delta}^2_1$ . If we do not assume $\sf{DC}$ , the notion of properness becomes obscure as pointed out by Asper\'{o} and Karagila. Using the notion of strong properness similar to the one introduced by Bagaria and Bosch, we show under $\sf{ZF} + \sf{DC}_{\mathbb{R}}$ without using $\sf{DC}$ that every set of reals is $I$ -regular for any $\sigma$ -ideal $I$ on the Baire space $\omega^{\omega}$ such that $\mathbb{P}_I$ is strongly proper assuming every set of reals is $\infty$ -Borel and there is no $\omega_1$ -sequence of distinct reals. In particular, the same conclusion holds in a Solovay model.

Keywords

i-regularity determinacy infinity-borel sets sigma-ideal proper forcing solovay model set of reals

Cite

@article{arxiv.2108.06632,
  title  = {$I$-regularity, determinacy, and $\infty$-Borel sets of reals},
  author = {Daisuke Ikegami},
  journal= {arXiv preprint arXiv:2108.06632},
  year   = {2021}
}

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

Abstract

Keywords

Cite

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