A constructive interpretation of the full set theory

E-ISSN： 1943-5886|52|1|172-201

ISSN： 0022-4812

Source： The Journal of Symbolic Logic, Vol.52, Iss.1, 1987-03, pp. : 172-201

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Abstract

The interpretation of the ZF set theory reported in this paper is, actually, part of a wider effort, namely, a new approach to the foundation of mathematics, which is referred to as The Cybernetic Foundation. A detailed exposition of the Cybernetic Foundation will be published elsewhere. Our approach leads to a full acceptance of the formalism of the classical set theory, but interprets it using only the idea of potential, but not actual (completed) infinity, and dealing only with finite objects that can actually be constructed. Thus we have a finitist proof of the consistency of ZF. This becomes possible because we set forth a metatheory of mathematics which goes beyond the classical logic and set theory and, of course, cannot be formalized in ZF, yet yields proofs which are as convincing—at least, from the author's viewpoint—as any mathematical proof can be.