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

Vladimir Kanovei and Vassily Lyubetsky
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Vladimir Kanovei: Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute), 127051 Moscow, Russia
Vassily Lyubetsky: Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute), 127051 Moscow, Russia

Mathematics, 2022, vol. 10, issue 3, 1-16

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.

Keywords: forcing; HOD sets; countable sets; nontypical sets (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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