Hostname: page-component-77c89778f8-sh8wx Total loading time: 0 Render date: 2024-07-24T19:01:44.177Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1] Dawson, J. W., Completing the Gödel-Zermelo correspondence, Historia Mathematica, vol. 12 (1985), pp. 66–70.CrossRefGoogle Scholar
[2] Dawson, J. W., Logical dilemmas. The life and work of Kurt Gödel, A. K. Peters, Wellesley, Massachusetts, 1997.Google Scholar
[3] Fraenkel, A., Einleitung in die Mengenlehre, Springer, Berlin, 1919.CrossRefGoogle Scholar
[4] Fraenkel, A., Über den Begriff “definit” und die Unabhängigkeit des Auswahlaxioms, Die Preussische Akademie der Wissenschaften. Sitzungsberichte. Physikalisch-Mathematische Klasse, (1922), pp. 253–257.Google Scholar
[5] Fraenkel, A., Zehn Vorlesungen über die Grundlegung der Mengenlehre, Teubner, Leipzig, 1927.CrossRefGoogle Scholar
[6] Grattan-Guinness, I., In memoriam Kurt Gödel: His 1931 correspondence with Zermelo on his incompletability theorem, Historia Mathematica, vol. 6 (1979), pp. 294–304.CrossRefGoogle Scholar
[7] Kanamori, A., The higher infinite. Large cardinals in set theory from their beginnings, Springer, Berlin, 1994.Google Scholar
[8] Moore, G. H., Beyond first-order logic: the historical interplay between mathematical logic and axiomatic set theory, History and Philosophy of Logic, vol. 1 (1980), pp. 95–137.CrossRefGoogle Scholar
[9] Rang, B. and Thomas, W., Zermelo's discovery of the ‘Russell’ paradox, Historia Mathematica, vol. 8 (1981), pp. 15–22.CrossRefGoogle Scholar
[10] Skolem, Th., Einige Bemerkungen zur axiomatischen Begründung der Mengenlehre, Proceedings of the 5th Scandinavian Mathematics Congress, Helsinki 1922, Akademiska Bokhandeln, Helsinki, 1923, pp. 217–232.Google Scholar
[11] Skolem, Th., Über einige Grundlagenfragen der Mathematik, Norsk Vid.-Akad.Oslo.Mat. nat. Klasse, vol. 4 (1929), pp. 1–29.Google Scholar
[12] Skolem, Th., Über die Grundlagendiskussionen in der Mathematik, Proceedings of the 7th Scandinavian Mathematics Congress, Oslo 1929, Broggers, Oslo, 1930, pp. 3–21.Google Scholar
[13] Taylor, R. G., Zermelo, reductionism and the philosophy of mathematics, Notre Dame Journal of Formal Logic, vol. 34 (1993), pp. 539–563.CrossRefGoogle Scholar
[14] Heijenoort, J. van, From Frege to Gödel, Harvard University Press, Cambridge, Massachusetts, 1967.Google Scholar
[15] Neumann, J. von, Eine Axiomatisierung der Mengenlehre, Journal für die Reine und Angewandte Mathematik, vol. 154 (1925), pp. 219–240.CrossRefGoogle Scholar
[16] Zermelo, E., Untersuchungen über die Grundlagen der Mathematik I, Mathematische Annalen, vol. 65 (1908), pp. 261–281.CrossRefGoogle Scholar
[17] Zermelo, E., Über den Begriff der Definitheit in der Axiomatik, Fundamenta Mathematicae, vol. 14 (1929), pp. 339–344.CrossRefGoogle Scholar
[18] Zermelo, E., Vortrags-Themata für Warschau 1929, manuscript in the Nachlass; partly reproduced in [8], 1929.Google Scholar
[19] Zermelo, E., Über Grenzzahlen und Mengenbereiche. Neue Untersuchungen über die Grundlagen der Mengenlehre, Fundamenta Mathematicae, vol. 16 (1930), pp. 29–47.CrossRefGoogle Scholar
[20] Zermelo, E., Bericht an die Notgemeinschaft der Deutschen Wissenschaft über meine Forschungen betreffend die Grundlagen der Mathematik, manuscript in the Nachlass; reproduced in [8], 1931?Google Scholar
[21] Zermelo, E., Über Stufen der Quantifikation und die Logik des Unendlichen, Jahresbericht der Deutschen Mathematiker-Vereinigung, vol. 41 (1932), pp. 85–92.Google Scholar
[22] Zermelo, E., Grundlagen einer allgemeinen Theorie dermathematischen Satzsysteme (Erste Mitteilung), Fundamenta Mathematicae, vol. 25 (1935), pp. 136–146.CrossRefGoogle Scholar