Predicate logics of constructive arithmetical theories

Abstract

In this paper, we show that the predicate logics of consistent extensions of Heyting’s Arithmetic plus Church’s Thesis with uniqueness condition are complete Π⁰₂. Similarly, we show that the predicate logic of HA^*, i.e. Heyting’s Arithmetic plus the Completeness Principle (for HA^*) is complete Π⁰₂. These results extend the known results due to Valery Plisko. To prove the results we adapt Plisko’s method to use Tennenbaum’s Theorem to prove ‘categoricity of interpretations’ under certain assumptions.