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Is Any Set Theory True?

Published online by Cambridge University Press:  14 March 2022

Joseph S. Ullian
Joseph S. Ullian*
Affiliation:
Washington University
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Abstract

This paper draws its title from the recent symposium of which it was part; it attempts to respond to the question raised by that title, taking current work in set theory into account. To this end the paper contrasts set theory with number theory, examines a severe brand of set-theoretic realism that is suggested by a passage from Gödel, and sketches a first-order way of looking at the results about competing extensions of Zermelo-Fraenkel set theory. A formalistic sentiment may be detectable in some portions of the paper.

Type
Research Article
Information
Philosophy of Science , Volume 36 , Issue 3 , September 1969 , pp. 271 - 279
Copyright
Copyright © 1969 by The Philosophy of Science Association

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Footnotes

1

This is the text of a paper read at a joint symposium of the Association for Symbolic Logic and the American Philosophical Association on 27 December, 1967, in Boston. The paper bears the symposium's title; the other symposiasts were Donald A. Martin and Saul Kripke. I am very greatly indebted to B. S. Dreben, H. Putnam, and W. V. Quine, each of whom was a rich and generous source of insight and stimulation for me. But this is not to accuse any of them of sympathy with my position.

References

[1] Cohen, P. J., Set Theory and the Continuum Hypothesis, Benjamin, New York, 1966.Google Scholar
[2] Gödel, K., “What is Cantor's Continuum Problem?” in Benacerraf, P. and Putnam, H. (eds.), Philosophy of Mathematics, Prentice-Hall, Englewood Cliffs, New Jersey, 1964.Google Scholar
[3] Putnam, H., “The Thesis that Mathematics is Logic,” in Schoenman, R. (ed.), Bertrand Russell, Philosopher of the Century, Little, Brown and Company, Boston, 1967.Google Scholar
[4] Quine, W. V., Set Theory and Its Logic, Belknap Press, Cambridge, 1963.CrossRefGoogle Scholar
[5] Skolem, T., “Some Remarks on Axiomatized Set Theory,” in van Heijenoort, J. (ed.), From Frege to Gödel, Harvard Press, Cambridge, 1967.Google Scholar