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An intuitionistic version of Zermelo's proof that every choice set can be well-ordered

Published online by Cambridge University Press:  12 March 2014

J. Todd Wilson
J. Todd Wilson*
Affiliation:
Department of Computer Science, California State University, Fresno, Fresno, CA 93740, E-mail: [email protected]
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Abstract

We give a proof, valid in any elementary topos, of the theorem of Zermelo that any set possessing a choice function for its set of inhabited subsets can be well-ordered. Our proof is considerably simpler than existing proofs in the literature and moreover can be seen as a direct generalization of Zermelo's own 1908 proof of his theorem.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 66 , Issue 3 , September 2001 , pp. 1121 - 1126
Copyright
Copyright © Association for Symbolic Logic 2001

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References

REFERENCES

[1]Aczel, P., Every choice set can be well-ordered: An elementary proof in intuitionistic set theory, Atti del Congresso Temi e prospettive della logica e della filosofia della scienze contemporanee (Cesena 7-10 gennaio, 1987), CLUEB, Bologna, 1988, pp. 13–22.Google Scholar
[2]Bell, J. L., Toposes and local set theories: An introduction, Oxford Logic Guides, vol. 14, Oxford University Press, Oxford, 1988.Google Scholar
[3]Blass, A., Well-ordering and induction in intuitionistic logic and topoi, Mathematical logic and theoretical computer science (College Park, Md., 1984–1985), Dekker, New York, 1987, pp. 29–48.Google Scholar
[4]Freyd, P., Choice and well-ordering, Annals of Pure and Applied Logic, vol. 35 (1987), pp. 149–166.CrossRefGoogle Scholar
[5]van Heltenoort, J., From Frege to Gödel: A sourcebook in mathematical logic, 1897–1931, Harvard University Press, Cambridge, 1967.Google Scholar
[6]Smullyan, R. M. and Fitting, M., Set theory and the continuum problem, Oxford Logic Guides, vol. 34, Oxford University Press, Oxford, 1996.Google Scholar
[7]Wilson, J. T., An intuitionistic fixed-point theorem, preprint, 1999.Google Scholar
[8]Zermelo, E., Beweis, daß jede Menge wohlgeordnet werden kann, Mathematische Annalen, vol. 59 (1904), pp. 514–516, English translation in [5], pp. 139–141.CrossRefGoogle Scholar
[9]Zermelo, E., Neuer Beweis für die Möglichkeit einer Wohlordnung, Mathematische Annalen, vol. 65 (1908), pp. 107–128, English translation in [5], pp. 183–198.CrossRefGoogle Scholar