On Russell typicality in set theory

According to Tzouvaras, a set is nontypical in the Russell sense if it belongs countable ordinal definable set. The class H mathvariant="bold">N mathvariant="bold">T \mathbf {HNT} of all hereditarily sets satisfies axioms Z F"> mathvariant="bold">Z mathvariant="bold">F {ZF} and double inclusion O D subset-of-or-equal-to T V"> mathvariant="bold">O mathvariant="bold">D ⊆ mathvariant="bold">V {HOD}\subseteq \mathbf {HNT}\subseteq {V} holds. Several questions about nature such sets, recently proposed by are solved this paper. In particular, model F C"> mathvariant="bold">C {ZFC} presented which subset-of-above-not-equals ⫋<!-- ⫋ {HOD}\subsetneqq {HNT}\subsetneqq </inline-formula>, another does not satisfy axiom choice.