On Russell typicality in set theory

Kanovei, Vladimir; Lyubetsky, Vassily

doi:10.1090/proc/16232

On Russell typicality in set theory
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by Vladimir Kanovei and Vassily Lyubetsky

Proc. Amer. Math. Soc. 151 (2023), 2201-2210

DOI: https://doi.org/10.1090/proc/16232

Published electronically: February 28, 2023

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Abstract:

According to Tzouvaras, a set is nontypical in the Russell sense if it belongs to a countable ordinal definable set. The class $\mathbf {HNT}$ of all hereditarily nontypical sets satisfies all axioms of $\mathbf {ZF}$ and the double inclusion $\mathbf {HOD}\subseteq \mathbf {HNT}\subseteq \mathbf {V}$ holds. Several questions about the nature of such sets, recently proposed by Tzouvaras, are solved in this paper. In particular, a model of $\mathbf {ZFC}$ is presented in which $\mathbf {HOD}\subsetneqq \mathbf {HNT}\subsetneqq \mathbf {V}$, and another model of $\mathbf {ZFC}$ in which $\mathbf {HNT}$ does not satisfy the axiom of choice.

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