We adjust the notion of typicality 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 ${ \rm NT}$ 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 $V_\zeta$ of $V$. This strengthening leads to defining ${ \rm NT}$ as the class of sets that belong to some countable ordinal definable set. It follows that ${ \rm OD}\subseteq { \rm NT}$ and hence ${ \rm HOD}\subseteq { \rm HNT}$. It is proved that the class ${ \rm HNT}$ of hereditarily nontypical sets is an inner model of ${ \rm ZF}$. Moreover the (relative) consistency of $V\neq { \rm NT}$ 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 ${ \rm HOD}\neq { \rm HNT}$ is also relatively consistent. In particular it is consistent that ${ \cal P}(\omega)\cap { \rm OD}\subsetneq{ \cal P}(\omega)\cap { \rm NT}$. However many questions remain open, among them the consistency of ${ \rm HOD}\neq { \rm HNT}\neq V$, ${ \rm HOD}={ \rm HNT}\neq V$ and ${ \rm HOD}\neq { \rm HNT}= V$.
Categories: math.LO