On the relative strengths of fragments of collection

Abstract

Let M be the basic set theory that consists of the axioms of extensionality, emptyset, pair, union, powerset, infinity, transitive containment, Δ0-separation and set foundation. This paper studies the relative strength of set theories obtained by adding fragments of the set-theoretic collection scheme to M. We focus on two common parameterisations of the collection: -collection, which is the usual collection scheme restricted to -formulae, and strong -collection, which is equivalent to -collection plus -separation. The main result of this paper shows that for all , 1. M + proves that there exists a transitive model of Zermelo Set Theory plus -collection, 2. the theory M + is -conservative over the theory M + strong . It is also shown that (2) holds for when the Axiom of Choice is included in the base theory. The final section indicates how the proofs of (1) and (2) can be modified to obtain analogues of these results for theories obtained by adding fragments of collection to a base theory (Kripke-Platek Set Theory with Infinity plus V=L) that does not include the powerset axiom.