A Generic Model in Which the Russell-Nontypical Sets Satisfy ZFC Strictly between HOD and the Universe

Abstract

The notion of ordinal definability and the related notions of ordinal definable sets (class OD) and hereditarily ordinal definable sets (class HOD) belong to the key concepts of modern set theory. Recent studies have discovered more general types of sets, still based on the notion of ordinal definability, but in a more blurry way. In particular, Tzouvaras has recently introduced the notion of sets nontypical in the Russell sense, so that a set x is nontypical if it belongs to a countable ordinal definable set. Tzouvaras demonstrated that the class HNT of all hereditarily nontypical sets satisfies all axioms of ZF and satisfies HOD⊆HNT. In view of this, Tzouvaras proposed a problem—to find out whether the class HNT can be separated from HOD by the strict inclusion HOD⫋HNT, and whether it can also be separated from the universe V of all sets by the strict inclusion HNT⫋V, in suitable set theoretic models. Solving this problem, a generic extension L[a,x] of the Gödel-constructible universe L, by two reals a,x, is presented in this paper, in which the relation L=HOD⫋L[a]=HNT⫋L[a,x]=V is fulfilled, so that HNT is a model of ZFC strictly between HOD and the universe. Our result proves that the class HNT is really a new rich class of sets, which does not necessarily coincide with either the well-known class HOD or the whole universe V. This opens new possibilities in the ongoing study of the consistency and independence problems in modern set theory.