On Russell typicality in Set Theory

Abstract

By Tzouvaras, 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. Several questions about the nature of such sets, recently proposed by Tzouvaras, are solved in this paper. In particular, a model of ZFC is presented in which HOD$\subsetneqq$HNT$\subsetneqq$V, and another model of ZFC, in which HNT does not satisfy the axiom of choice.