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.References
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Bibliographic Information
- Vladimir Kanovei
- Affiliation: Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute), Moscow, Russia
- MR Author ID: 97930
- ORCID: 0000-0001-7415-9784
- Email: kanovei@iitp.ru
- Vassily Lyubetsky
- Affiliation: Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute), Moscow, Russia
- MR Author ID: 209834
- ORCID: 0000-0002-3739-9161
- Email: lyubetsk@iitp.ru
- Received by editor(s): November 15, 2021
- Received by editor(s) in revised form: August 20, 2022
- Published electronically: February 28, 2023
- Additional Notes: The authors were partially supported by RFBR grant 20-01-00670.
- Communicated by: Vera Fischer
- © Copyright 2023 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 151 (2023), 2201-2210
- MSC (2020): Primary 03E35; Secondary 03E15
- DOI: https://doi.org/10.1090/proc/16232
- MathSciNet review: 4556211