Typicality à la Russell in Set Theory

Abstract

We adjust the notion of typicality that originated with Russell, which was introduced and studied in a previous paper for general first-order structures, to make it expressible in the language of set theory. The adopted definition of the class $\mathrm{NT}$ of nontypical sets comes out as a natural strengthening of Russell’s initial definition, which employs properties of small (minority) extensions, when the latter are restricted to the various levels $V_{\mathit{\zeta }}$ of V. This strengthening leads to defining $\mathrm{NT}$ as the class of sets that belong to some countable ordinal definable set. It follows that $\mathrm{OD}\subseteq \mathrm{NT}$, and hence $\mathrm{HOD}\subseteq \mathrm{HNT}$. It is proved that the class $\mathrm{HNT}$ of hereditarily nontypical sets is an inner model of $\mathrm{ZF}$. Moreover, the (relative) consistency of $V\ne \mathrm{NT}$ is established, by showing that in many forcing extensions $M[G]$ the generic set G is a typical element of $M[G]$, a fact which is fully in accord with the intuitive meaning of typicality. In particular, it is consistent that there exist continuum many typical reals. In addition, it follows from a result of Kanovei and Lyubetsky that $\mathrm{HOD}\ne \mathrm{HNT}$ is also relatively consistent. In particular, it is consistent that $\mathcal{P}(\mathit{\omega })\cap \mathrm{OD}⊊\mathcal{P}(\mathit{\omega })\cap \mathrm{NT}$. However, many questions remain open, among them the consistency of $\mathrm{HOD}\ne \mathrm{HNT}\ne V$, $\mathrm{HOD}=\mathrm{HNT}\ne V$, and $\mathrm{HOD}\ne \mathrm{HNT}=V$.