A product forcing model in which the Russell-nontypical sets satisfy ZFC strictly between HOD and the universe

Vladimir Kanovei; Vassily Lyubetsky

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arXiv · 2111.13491

A product forcing model in which the Russell-nontypical sets satisfy ZFC strictly between HOD and the universe

Vladimir Kanovei·Vassily Lyubetsky

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Abstract

A set is nontypical in the Russell sense, if it belongs to a countable ordinal definable set. The class HNT of all hereditarily nontypical sets satisfies all axioms of ZF and the double inclusion HOD $\subseteq$ HNT $\subseteq$ V holds. Solving a problem recently proposed by Tzouvaras, a generic extension L$[a,x]$ of L, by two reals $a,x$, is presented in which L=HOD $\subsetneqq$ L$[a]$=HNT $\subsetneqq$ V=L$[a,x]$, so that HNT is a model of ZFC strictly between HOD and the universe.

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BibTeX RIS

Vladimir Kanovei, Vassily Lyubetsky. 2021-11-26. A product forcing model in which the Russell-nontypical sets satisfy ZFC strictly between HOD and the universe. https://arxiv.org/abs/2111.13491

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