ELEMENTARY CONSTRUCTIVE OPERATIONAL SET THEORY A. CANTINI, L. CROSILLA Dedicated to Prof. Wolfram Pohlers Abstract. We introduce an operational set theory in the style of [5] and [16]. The theory we develop here is a theory of constructive sets and operations. One motivation behind constructive operational set theory is to merge a con- structive notion of set ([1], [2]) with some aspects which are typical of explicit mathematics [14]. In particular, one has non-extensional operations (or rules) alongside extensional constructive sets. Operations are in general partial and a limited form of self–application is permitted. The system we introduce here is a fully explicit, finitely axiomatised system of constructive sets and operations, which is shown to be as strong as HA. 1. Introduction This article is a follow-up of [9], where a constructive set theory with operations was introduced. Constructive operational set theory (COST) is a constructive theory of sets and operations which has similarities with Feferman’s (classical) Operational Set Theory ([16], [17], [20], [21], [22]) and Beeson’s Intuitionistic set theory with rules [5]. In this article a fully explicit fragment, called EST, of COST is singled out. This system is finitely axiomatized and is shown to be proof–theoretically as strong as Peano Arithmetic, PA(section 5). One motivation behind constructive operational set theory is to merge a con- structive notion of set ([25], [1], [2]) with some aspects which are typical of explicit mathematics [14]. In particular, one has non-extensional operations (or rules) along- side extensional constructive sets. Operations are in general partial and a limited form of self–application is permitted. The informal concept of rule plays a prominent role in constructive mathemat- ics. Both Feferman and Beeson have repeatedly called attention to the distinction between rules and set–theoretic functions (see e.g. [15], [3]). There are several ex- amples of intuitive rules which can not be represented by the set–theoretic concept of function. For example the operation of pair, which given two sets a and b enables us to form a new set, the set–theoretic pair of a and b. In operational set theory we have primitive operations corresponding to some set–theoretic rules, among which that of pair. In a sense, rules can be regarded as generalized algorithms or abstract rules. Without entering a detailed conceptual analysis of the notion of rule, we simply adopt the view that rules are represented by sets, and that it makes sense to apply a set c ‘qua rule ’ to another set b as input; and this possibly provides a result, The research is part of a project supported by PRIN 2006 (Dimostrazioni, Operazioni, Insiemi). The second author gratefully acknowledges a grant by the John Templeton Foundation. She would also like to thank the School of Mathematics, University of Leeds, for the hospitality. 1