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

TL;DR

This paper constructs a model of set theory where hereditarily nontypical sets satisfy ZFC and lie strictly between HOD and the universe, solving a recent open problem.

Contribution

It introduces a new model of set theory where HNT satisfies ZFC and is strictly between HOD and the universe, addressing a problem posed by Tzouvaras.

Findings

HNT satisfies all axioms of ZF.

HOD is strictly contained in HNT, which is strictly contained in V.

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