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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.

Information

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

REFERENCES

Brouwer, L. E. J. [1949], Consciousness, philosophy, and mathematics, Library of the tenth international congress of philosophy (Amsterdam, 1948), vol. I, North-Holland, Amsterdam, pp. 1235–1249; reprinted in his Collected works , vol. 1 (Heyting, A., editor), North-Holland, Amsterdam, 1975, pp. 480–494.Google Scholar
Brouwer, L. E. J. [1952], Historical background, principles and methods of intuitionism, South African Journal of Science, vol. 49, pp. 139–146; reprinted in his Collected works , vol. 1 (Heyting, A., editor), North-Holland, Amsterdam, 1975, pp. 508–515.Google Scholar
Cantor, G. [1895–1897], Beiträge zur Begründungen der transfiniten Mengenlehre. I, II, Mathematische Annakn, vol. 46, pp. 485–512, and vol. 49, pp. 207–246; English translation (by P. E. B. Jourdain), Open Court, Chicago, Illinois, 1915; reprint, Dover, New York, 1952.CrossRefGoogle Scholar
FBL [1967]: Fraenkel, A. A., Bar-Hillel, Y., and Lévy, A., Foundations of set theory, 2nd rev. ed., North-Holland, Amsterdam.Google Scholar
Gödel, Kurt [1940], The consistency of the axiom of choice and the generalized continuum hypothesis with the axioms of set theory, Annals of Mathematics Studies, no. 3, Princeton University Press, Princeton, New Jersey.Google Scholar
Kleene, S. C. [1952], Introduction to metamathematics, Van Nostrand, New York. (See especially pp. 332–340.)Google Scholar
Kreisel, G. [1967], Informal rigour and completeness proofs, Problems in the philosophy of mathematics (proceedings of the international colloquium in the philosophy of science, London 1965), vol. 1 (Lakatos, I., editor), North-Holland, Amsterdam, pp. 138–186.Google Scholar
Kripke, S. [1975], Outline of a theory of truth, Journal of Philosophy, vol. 72, pp. 690–716.CrossRefGoogle Scholar
Martino, E. [1982], Creative subject and bar theorem, The L. E. J. Brouwer centenary symposium (Troelstra, A. S. and van Dalen, D., editors), North-Holland, Amsterdam, pp. 311–318.Google Scholar
Orlov, Yuri F. [1978], Wave calculus based on wave logic, International Journal of Theoretical Physics, vol. 17, pp. 585–598.CrossRefGoogle Scholar
Orlov, Yuri F. [1982], The wave logic of consciousness: a hypothesis, International Journal of Theoretical Physics, vol. 21, pp. 37–53.CrossRefGoogle Scholar
Rogers, H. [1967], Theory of recursive functions and effective computability, McGraw-Hill, New York.Google Scholar
Tarski, A. [1933], The concept of truth in the languages of deductive sciences, Prace Towarzystwa Naukowego Warszawskiego, Wydzial III, no. 34 (in Polish); German translation, Studia Philosophica , vol. 1 (1935), pp. 261–405; English translation in his Logic, semantics, metamathematics. Papers from 1923 to 1938 , Clarendon Press, Oxford, 1956, pp. 152–278.Google Scholar
Turchin, V. F. [1980], The language Refal, Courant Computer Science Report no. 20, New York University, New York.Google Scholar
Zermelo, E. [1890], Untersuchungen über die Grundlagen der Mengenlehre, Mathematische Annalen, vol. 65, pp. 261–281; English translation in From Frege to Gödel: a source book in mathematical logic (van Heijenoort, J., editor), Harvard University Press, Cambridge, Massachusetts, 1967, pp. 200–215.CrossRefGoogle Scholar