On Russell typicality in set theory

Abstract

According to Tzouvaras, a set is nontypical in the Russell sense if it belongs to a countable ordinal definable set. The class H N T \mathbf {HNT} of all hereditarily nontypical sets satisfies all axioms of Z F \mathbf {ZF} and the double inclusion H O D ⊆ H N T ⊆ V \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 Z F C \mathbf {ZFC} is presented in which H O D ⫋ H N T ⫋ V \mathbf {HOD}\subsetneqq \mathbf {HNT}\subsetneqq \mathbf {V} , and another model of Z F C \mathbf {ZFC} in which H N T \mathbf {HNT} does not satisfy the axiom of choice.