Multi-sorted version of second order arithmetic

Abstract

This paper describes axiomatic theories SA and SAR, which are versions of second order arithmetic with countably many sorts for sets of natural numbers. The theories are intended to be applied in reverse mathematics because their multi-sorted language allows to express some mathematical statements in more natural form than in the standard second order arithmetic. We study metamathematical properties of the theories SA, SAR and their fragments. We show that SA is mutually interpretable with the theory of arithmetical truth PATr obtained from the Peano arithmetic by adding infinitely many truth predicates. Corresponding fragments of SA and PATr are also mutually interpretable. We compare the proof-theoretical strengths of the fragments; in particular, we show that each fragment SAs with sorts <=s is weaker than next fragment SAs+1.