Transfinite Sequences of Constructive Predicate Logics

Abstract

The predicate logic of recursive realizability was carefully studied by the author in the 1970s. In particular, it was proved that the predicate realizability logic changes if the basic language of formal arithmetic is replaced by its extension with the truth predicate. In the paper this result is generalized by considering similar extensions of the arithmetical language by transfinite induction up to an arbitrary constructive ordinal. Namely, for every constructive ordinal α, a transfinite sequence of extensions LAβ (β≤α) of the first-order language of formal arithmetic is considered. Constructive semantics for these languages is defined in terms of recursive realizability. Variants of LAβ-realizability for the predicate formulas are considered and corresponding predicate logics are studied.