Conservativity of transitive closure over weak constructive operational set theory

Abstract

Constructive set theoryà la Myhill-Aczel has been extended in [10,11] to incorporate a notion of (partial, non-extensional) operation. Constructive operational set theory is a constructive and predicative analogue of Beeson's Inuitionistic set theory with rules and of Feferman's Operational set theory [4,15,16,17,18]. This paper is concerned with an extension of constructive operational set theory [11] by a uniform operation of Transitive Closure, τ. Given a set a, τ produces its transitive closure τa. We show that the theory ESTE of [11] augmented by τ is still conservative over Peano Arithmetic.