Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-24T22:32:57.588Z Has data issue: false hasContentIssue false
  • English
  • Français

On a Theory Objects Based on a Single Axiom Scheme

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.

There are some fundamental mathematical theories, such as the Fraenkel set-theory and the Bernays-Gödel set-theory, in which, I believe, all the actually important formal theories of mathematics can be embedded. Formal theories come into existence by being shown their consistency. As far as this is admitted, not all the axioms of set theory are necessary for a fundamental mathematical theory. The fundierung axiom is proved consistent by v. Neumann, the axiom of extensionality is proved consistent by Gandy, and even the axiom of choice is proved consistent by Göldel. Although it is not evident that a set-theory does not cease from being a fundamental theory of mathematics after abandoning these axioms all at once, the theory must be enough for being a fundamental theory of mathematics without some of them.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 26 , February 1966 , pp. 13 - 30
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1966

References

[1] Bernays, P. [1], A system of axiomatic set theory I-VII, J. Symb. Log,; I, vol. 2 (1937), 65–77; II, vol. 6 (1941), 1–17; III, vol. 7 (1942), 65–89; IV, vol. 7 (1942), 133–145; V, vol. 8 (1943), 89–106; VI, vol. 13 (1948), 65–79; VII, vol. 19 (1954), 81–96.CrossRefGoogle Scholar
[2] Fraenkel, A., [1] Untersuchungen über die Grundlagen der Mengenlehre, Math. Ztschr., vol. 22 (1925), 250–273.Google Scholar
[3] Fraenkel, A., [2] Einleitung in die Mengenlehre, 3rd. ed., Berlin (1928).CrossRefGoogle Scholar
[4] Gandy, R. O., [1] On the axiom of extensionality I, II, J. Symb. Log.; I, vol. 21 (1956), 36–48; II, vol. 24 (1959), 287–300.Google Scholar
[5] Gödel, K., [1] Tfye consistency of the axiom of choice and of the generalized continuum hypothesis with the axioms of set-theory, Annals of Math. Studies, No. 3, Princeton (1940).Google Scholar
[6] Neumann, v., , J., [1] Die Axiomatisierung der Mengenlehre, Math. Ztschr., vol. 21 (1928), 669–752.CrossRefGoogle Scholar
[7] Neumann, v., , J., [2] Über eine Widerspruchsfreiheitsfrage in der axiomatischen Mengenlehre, J. f. reine u. angw. Math., vol. 160 (1929), 227–241.Google Scholar
[8] Ono, K., [1] A set theory founded on unique generating principle, Nagoya Math. J., vol. 12 (1957), 151–159.Google Scholar
[9] Ono, K., [2] A theory of mathematical objects as a prototype of set theory, Nagoya Math. J., vol. 20 (1962), 105–168.Google Scholar
[10] Ono, K., [3] On a practical way of describing formal deductions, Nagoya Math. J., vol. 21 (1962), 115–121.Google Scholar
[11] Ono, K., [4] A stronger system of object theory as a prototype of set theory, Nagoya Math. J., vol. 22 (1963), 119–167.Google Scholar
[12] Quine, W. V. O., [1] New foundations for mathematical logic, Am. Math. Monthly, vol. 44 (1937), 70–80.Google Scholar
[13] Quine, W. V. O., [2] Mathematical logic, Cambridge Mass. (1940, revised ed. 1951).Google Scholar