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Zermelo and the Skolem Paradox

Published online by Cambridge University Press:  15 January 2014

Dirk Van Dalen  and
Heinz-Dieter Ebbinghaus
Dirk Van Dalen
Affiliation:
University of Utrecht, Department of Philosophy, P.O. Box 80.126, 3508 TC Utrecht, The Netherlands
Heinz-Dieter Ebbinghaus
Affiliation:
University of Freiburg, Institute Of Mathematical Logic, Eckerstrasse 1, 79104 Freiburg, Germany
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Extract

On October 4, 1937, Zermelo composed a small note entitled “Der Relativismus in der Mengenlehre und der sogenannte Skolemsche Satz”(“Relativism in Set Theory and the So-Called Theorem of Skolem”) in which he gives a refutation of “Skolem's paradox”, i.e., the fact that Zermelo-Fraenkel set theory—guaranteeing the existence of uncountably many sets—has a countable model. Compared with what he wished to disprove, the argument fails. However, at a second glance, it strongly documents his view of mathematics as based on a world of objects that could only be grasped adequately by infinitary means. So the refutation might serve as a final clue to his epistemological credo.

Whereas the Skolem paradox was to raise a lot of concern in the twenties and the early thirties, it seemed to have been settled by the time Zermelo wrote his paper, namely in favour of Skolem's approach, thus also accepting the noncategoricity and incompleteness of the first-order axiom systems. So the paper might be considered a late-comer in a community of logicians and set theorists who mainly followed finitary conceptions, in particular emphasizing the role of first-order logic (cf. [8]). However, Zermelo never shared this viewpoint: In his first letter to Gödel of September 21, 1931, (cf. [1]) he had written that the Skolem paradox rested on the erroneous assumption that every mathematically definable notion should be expressible by a finite combination of signs, whereas a reasonable metamathematics would only be possible after this “finitistic prejudice” would have been overcome, “a task I have made my particular duty”.

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 6 , Issue 2 , June 2000 , pp. 145 - 161
Copyright
Copyright © Association for Symbolic Logic 2000

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References

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