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Elementary constructive operational set theory. To appear in: Festschrift for Wolfram Pohlers, Ontos Verlag
"... Abstract. We introduce an operational set theory in the style of [5] and [17]. The theory we develop here is a theory of constructive sets and operations. One motivation behind constructive operational set theory is to merge a constructive notion of set ([1], [2]) with some aspects which are typical ..."
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Abstract. We introduce an operational set theory in the style of [5] and [17]. The theory we develop here is a theory of constructive sets and operations. One motivation behind constructive operational set theory is to merge a constructive notion of set ([1], [2]) with some aspects which are typical of explicit mathematics [14]. In particular, one has nonextensional 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.
Operations, sets and classes
"... Operational set theory, in the form described below, is an enterprise which consolidates classical set theory with some central concepts of Feferman’s explicit mathematics. It provides for a careful distinction between operations and settheoretic functions and as such reconciles set theory with nee ..."
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Operational set theory, in the form described below, is an enterprise which consolidates classical set theory with some central concepts of Feferman’s explicit mathematics. It provides for a careful distinction between operations and settheoretic functions and as such reconciles set theory with needs arising in constructive environments and even in those enhanced by computer science. In the following we consider, primarily from a prooftheoretic perspective, the theory OST and some of its most important extensions and determine their consistency strengths by exhibiting equivalent systems in the realm of traditional theories of sets and classes.