Hostname: page-component-8448b6f56d-cfpbc Total loading time: 0 Render date: 2024-04-23T17:08:32.364Z Has data issue: false hasContentIssue false
  • English
  • Français

The continuum hypothesis in intuitionism

Published online by Cambridge University Press:  12 March 2014

W. Gielen ,
H. de Swart  and
W. Veldman
W. Gielen
Affiliation:
Katholieke Universiteit, Nijmegen, The Netherlands
H. de Swart
Affiliation:
Katholieke Universiteit, Nijmegen, The Netherlands
W. Veldman
Affiliation:
Katholieke Universiteit, Nijmegen, The Netherlands
Get access Rights & Permissions [Opens in a new window]

Extract

Although Brouwer became famous for his vehement attacks upon classical logic and set theory, his work did not develop in a vacuum and strongly depended on that of Cantor.

His mind bent on shifting aside nonconstructive arguments, he tried to rebuild Cantor's edifice along new, intuitionistic lines. The continuum hypothesis, lying at the core of set theory, also confronted Brouwer, and he had to face the farthest conclusion Cantor had been able to reach in trying to solve it: every nondenumerable closed subset of the real line has the power of the continuum.

Brouwer's thinking about it seems to have been subject to some development. In 1914 we hear him saying: “Wir sahen oben dass das Cantorsche Haupttheorem für den Intuitionisten keines Beweises bedarf” (“As we saw above, for us, being intuitionists, Cantor's Main Theorem does not need a proof”) [3]. Nevertheless, five years later, he publishes an essay: Theorie der Punktmengen, which might be described as an attempt to reconstruct Cantor's reasonings in detail [4].

This attempt was not entirely successful, as Brouwer comes to admit in 1952, probably having lost, now, some of his youthful rashness [10]. So the question of what the constructive content of Cantor's Main Theorem is, still awaits an answer.

We do not think the answer we will give can be considered a conclusive one, but, in any case, it is a beginning.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 46 , Issue 1 , March 1981 , pp. 121 - 136
Copyright
Copyright © Association for Symbolic Logic 1981

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]Cantor, G., Über unendliche lineare Punktmannigfaltigkeiten, Mathematische Annalen, vol. 20 (1882), pp. 113–121; Mathematische Annalen, vol. 21 (1883). Also in G. Cantor, Gesammelte Abhandlungen, Springer-Verlag, Berlin, 1932.CrossRefGoogle Scholar
[CW] Brouwer, L. E. J., Collected works. Vol. I. Philosophy and foundations of mathematics, North-Holland, Amsterdam, 1975.Google Scholar
[2]Brouwer, L. E. J., Die moeglichen Mächtigkeiten, Atti del IV Congresso Internationale del Matematici, Bologna, 1908, Kraus Reprint Limited, Nendeln, Liechtenstein, 1969, pp. 569–571; [CW], pp. 102–104.Google Scholar
[3]Brouwer, L. E. J., Review of Schoenflies, A. und Hahn, H., Die Entwicklung der Mengenlehre und ihrer Anwendungen, Teubner, Leipzig und Berlin, 1913, Jahresbericht der Deutschen Mathematiker-Vereinigung, vol. 23 (1914), pp. 78–83 kursiv; [CW], pp. 139–144.Google Scholar
[4]Brouwer, L. E. J., Begründung der Mengenlehre unabhängig vom logischen Satz vom ausgeschlossenen Dritten. Zweiter Teil: Theorie der Punktmengen, Koninklijke Nederlandse Akademie van Wetenschappen. Verhandelingen, le sectie 12 (1919); no. 5, 43 p. [CW], pp. 191–221.Google Scholar
[5]Brouwer, L. E. J., Zur Begründung der intuitionistischen Mathematik. I. Mengen und Spezies. Ihre Vergleichung, Mathematische Annalen, vol. 93 (1925), pp. 244–257; [CW], pp. 301–314.CrossRefGoogle Scholar
[6]Brouwer, L. E. J., Zur Begrüindung der intuitionistischen Mathematik. III. Wohlordnung, Mathematische Annalen, vol. 96 (1927), pp. 451–488; [CW], pp. 352–389.CrossRefGoogle Scholar
[7]Brouwer, L. E. J., Beweiss dass der Begriff der Menge höherer Ordnung nicht als Grundbegriff der intuitionistischen Mathematik in Betracht kommt, Koninklijke Nederlandse Akademie van Wetenschappen. Proceedings, vol. 45 (1942), pp. 791–793; also Indagationes Mathematicae ex Actis Quibus Titulus, vol. 4 (1942), pp. 274–276; [CW], pp. 462–464.Google Scholar
[8]Brouwer, L. E. J., Consciousness, philosophy and mathematics, Proceedings of the 10th International Congress of Philosophy, North-Holland, Amsterdam, 1948, pp. 1235–1249; [CW], pp. 480–494.Google Scholar
[9]Brouwer, L. E. J., De non-aequivalentie van de constructieve en de negatieve orderelatie in het continuum, Koninklijke Nederlandse Akademie van Wetenschappen. Proceedings, vol. 52 (1949), pp. 122–124; also Indagationes Mathematicae ex Actis Quibus Titulus, vol. 11 (1949), pp. 89–90; [CW], pp. 495–496.Google Scholar
[10]Brouwer, L. E. J., Over accumulatiekernen van oneindige kernsoorten, Koninklijke Nederlandse Akademie van Wetenschappen. Proceedings. Series A, vol. 55 (1952), pp. 439–441; also Indagationes Mathematicae ex Actis Quibus Titulus, vol. 14 (1952), pp. 439–441; [CW], pp. 516–518.Google Scholar
[11]Hausdorff, F., Mengenlehre, 3rd edition, de Gruyter, Berlin, 1937.Google Scholar
[12]Lusin, N., Leçons sur les ensembles analytiques, Chelsea, New York, 1972.Google Scholar
[13]Kreisel, G., Analysis of the Cantor-Bendixson Theorem by means of the analytic hierarchy, Bulletin de l'Académie Polonaise des Sciences, Série des Sciences Mathématiques, Astronomiques et Physiques, VII, vol. 10 (1959), pp. 621–626.Google Scholar
[14]Kreisel, G., On weak completeness of intuitionistic predicate logic, this Journal, vol. 27 (1962), pp. 139–158.Google Scholar
[15]Kreisel, G., Informal rigour and completeness proofs, Problems in the philosophy of mathematics (Lakatos, I., editor), North-Holland, Amsterdam, 1967, pp. 138–186.CrossRefGoogle Scholar
[16]Kleene, S.C. and Vesley, R.E., The foundations of intuitionistic mathematics, especially in relation to recursive functions, North-Holland, Amsterdam, 1965.Google Scholar
[17]Troelstra, A.S., Choice sequences. A chapter of intuitionistic mathematics, Clarendon Press, Oxford, 1977.Google Scholar
[18]Hinman, P.G., Recursion-theoretic hierarchies, Springer-Verlag, Berlin, 1978.CrossRefGoogle Scholar
[19]Burgess, J., Brouwer and Souslin on transfinite cardinals, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik (to appear).Google Scholar
[20]Veldman, W., Investigations in intuitionistic hierarchy theory, (in preparation).Google Scholar