Validity in recursive structures was investigated by several authors. Kreisel  has shown that there exists a consistent sentence of classical predicate calculus (CPC) that does not possess a recursive model. The sentence is a conjunction of the axioms of a variant of Bernays set theory, including the axiom of infinity. The language contains additional constants besides ϵ. Later Kreisel  and Mostowski  presented a sentence (not possessing recursive models) which was a conjunction of axioms of a variant of Bernays set theory without the axiom of infinity but still with additional constants besides ϵ. Later Mostowski  improved the result by giving a sentence which can be demonstrated in Heyting arithmetic to be consistent and to have no recursive models. Rabin  obtained a simple proof that some sentence of set theory with the single nonlogical constant ϵ does not have any recursively enumerable models.
More generally, Mostowski  has shown that the set of all sentences valid in all RE models is not arithmetical and Vaught  improved this result by showing that it holds for a language of one binary relation. In fact, Vaught gives· a way of translating n-place relations to 2-place ones that preserves the RE characteristic of the model. For further results pertaining to recursive models see Vaught .