Notre Dame J. Formal Logic 63 (2), 185-196, (May 2022) DOI: 10.1215/00294527-2022-0011
KEYWORDS: Russell’s notion of typicality, typical and nontypical set of ZFC, ordinal definable set, 03E45, 03E47
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 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 of V. This strengthening leads to defining as the class of sets that belong to some countable ordinal definable set. It follows that , and hence . It is proved that the class of hereditarily nontypical sets is an inner model of . Moreover, the (relative) consistency of is established, by showing that in many forcing extensions the generic set G is a typical element of , 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 is also relatively consistent. In particular, it is consistent that . However, many questions remain open, among them the consistency of , , and .