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A constructive interpretation of the full set theory

Published online by Cambridge University Press:  12 March 2014

Valentin F. Turchin
Valentin F. Turchin*
Affiliation:
Department of Computer Sciences, The City College, City University of New York, New York, New York 10031
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Extract

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.

Our metatheory is based on the following two ideas. Firstly, we define the semantics of the mathematical language using the cybernetical concept of knowledge. According to this concept, to say that a cybernetic system (a human being, in particular) has some knowledge is to say that it has some models of reality. In the Cybernetic Foundation we consider mathematics as the art of constructing linguistic models of reality.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 52 , Issue 1 , March 1987 , pp. 172 - 201
Copyright
Copyright © Association for Symbolic Logic 1987

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Footnotes

1

Partly supported by NSF grants DCR-8007565 and DCR-8412986

References

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