On Russell typicality in Set Theory

TL;DR

This paper investigates the properties of hereditarily nontypical sets in set theory, demonstrating models where these sets differ from hereditarily ordinal definable sets and exploring their relation to the axiom of choice.

Contribution

It provides new models of ZFC where hereditarily nontypical sets differ from HOD and may violate the axiom of choice, advancing understanding of Russell typicality.

Findings

Constructed a ZFC model with HOD strictly contained in HNT and HNT strictly contained in V.

Presented a ZFC model where HNT does not satisfy the axiom of choice.

Solved several open questions about the nature of hereditarily nontypical sets.

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.