Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-23T23:40:01.414Z Has data issue: false hasContentIssue false
  • English
  • Français

A Set Theory Founded on Unique Generating Principle

Published online by Cambridge University Press:  22 January 2016

Katuzi Ono
Katuzi Ono*
Affiliation:
Mathematical Institute, Nagoya University
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The most important thing for a set theory seems to be that it can generate new mathematical objects, and I think that there must be an underlying principle, simple and unique, which unifies the acts of generating. The naive set theory has a unique generating principle, which defines, by any proposition on a variable x, the set of all x’s satisfying the proposition. Certainly, we must restrict this generating principle so as to exclude all contradictions it contains, without losing its essential rôle as logic of mathematics, and at the same time we would like to keep its uniqueness and simplicity.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 12 , December 1957 , pp. 151 - 159
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1957

References

[ 1 ] Zermelo, E., Untersuchungen ueber die Grundlagen der Mengenlehre, I, Math. Ann., vol. 65 (1908), pp. 261–281.CrossRefGoogle Scholar
[ 2 ] Fraenkel, A., Einleitung in die Mengenlehre, 1928 (3. Aufl.).Google Scholar
[ 3 ] Neumann, J. v., Die Axiomatisierung der Mengenlehre, Math. Z., vol. 27 (1928), pp. 669–752.Google Scholar
[ 4 ] Bernays, P., A system of axiomatic set theory, Part I-VII; Journ. of Symbolic Logic; I, vol. 2 (1937), pp. 65–77; II, vol. 6 (1941), pp. 1–17; III, vol. 7 (1942), pp. 65–89; IV, vol. 7 (1942), pp. 133–145; V, vol. 8 (1943), pp. 89–106; VI, vol. 13 (1948), pp. 65–79; VII, vol. 19 (1954), pp. 81–96.Google Scholar
[ 5 ] Bourbaki, N., Theorie des ensembles, Actualités Scientifiques et Industrielles, 846, 1939.Google Scholar
[ 6 ] Goedel, K., The consistency of the axiom of choice and of the generalized continuum hypothesis with the axioms of set theory, Annals of Mathematics Studies, No. 3, 1940.Google Scholar
[ 7 ] Ackermann, W., Zur Axiomatik der Mengenlehre, Math. Ann., vol. 131 (1956), pp. 336–347.Google Scholar
[ 8 ] Whitehead, A. N. and Russell, B., Principia mathematica, vol. 1, 1925 (2nd ed.).Google Scholar
[ 9 ] Quine, W. V., New foundation for mathematical logic, Amer. Math. Monthly, vol. 44 (1937), pp. 70–80.CrossRefGoogle Scholar
[10] Quine, W. V., Mathematical Logic, 1951 (revised edition), Cambridge.Google Scholar