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

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 $⫋$ L$[a]$=HNT $⫋$ V=L$[a,x]$, so that HNT is a model of ZFC strictly between HOD and the universe.

arXiv admin note: substantial text overlap with arXiv:2111.07654

References in corpus (1)

• Non-uniformizable sets with countable cross-sections on a given level of the projective hierarchy