The notion of ordinal definability and the related notions of ordinal definable sets (class
) and hereditarily ordinal definable sets (class
) belong to the key concepts of modern set theory. Recent studies have discovered more general types of sets, still
[...] Read more.
The notion of ordinal definability and the related notions of ordinal definable sets (class
) and hereditarily ordinal definable sets (class
) 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
of all hereditarily nontypical sets satisfies all axioms of
and satisfies
. In view of this, Tzouvaras proposed a problem—to find out whether the class
can be separated from
by the strict inclusion
, and whether it can also be separated from the universe
of all sets by the strict inclusion
, in suitable set theoretic models. Solving this problem, a generic extension
of the Gödel-constructible universe
, by two reals
, is presented in this paper, in which the relation
is fulfilled, so that
is a model of
strictly between
and the universe. Our result proves that the class
is really a new rich class of sets, which does not necessarily coincide with either the well-known class
or the whole universe
. This opens new possibilities in the ongoing study of the consistency and independence problems in modern set theory.
Full article
