Hostname: page-component-848d4c4894-m9kch Total loading time: 0 Render date: 2024-06-08T11:31:09.322Z Has data issue: false hasContentIssue false

A Certain Kind of Formal Theories

Published online by Cambridge University Press:  22 January 2016

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.

A common feature of formal theories is that each theory has its own system of axioms described in terms of some symbols for its primitive notions together with logical symbols. Each of these theories is developed by deduction from its axiom system in a certain logical system which is usually the classical logic of the first order.

Type
Research Article
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1965

References

[1] Bernays, P. A system of axiomatic set theory, Part I-VII; Symbolic Logic; I, vol. 2 (1937), pp. 6577; II, vol. 6 (1941), pp. 117; III, vol. 7 (1942), pp. 6565; IV, vol. 7 (1942), pp. 133133; V, vol. 8 (1943), pp. 8989; VI, vol. 13 (1948), pp. 6565; VII, vol. 19 (1954), pp. 8196.Google Scholar
[2] Gentzen, G. Untersuchungen über das logischen Schliessen, Math. Z., 39, pp. 176210, 405431 (1935).Google Scholar
[3] Glivenko, V. Sur quelques points de la logique de M. Brouwer, Academie Royale de Belgique, Bull, de la classe des sciences, ser. 5, 15 (1929), pp. 183183.Google Scholar
[4] Gödei, K. Über formal unentscheidbare Sätze der Principia Mathematica und ver-wandter Systeme I. Monatsch. f. Math. u. Phys., vol. 38 (1931), pp. 173173.Google Scholar
[5] Gödei, K. The 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.Google Scholar
[6] Johansson, I. Der Minimalkalkül, ein reduzierter intutionistischer Formalismus, Compositio Mathematicae, vol. 4 (1936), pp. 119119.Google Scholar
[7] Kuroda, S. Intuitionistische Untersuchungen der Formalistischen Logik, Nagoya. Math. J., vol. 2 (1951), pp. 3535.Google Scholar
[8] Nishimura, T., On Gödel’s theorem, J. Math. Soc. Jap., vol. 13 (1961), pp. 11.Google Scholar
[9] Ono, K., A theory of mathematical objects as a prototype of set theory, Nagoya Math. J., vol. 20 (1962), pp. 106106.CrossRefGoogle Scholar
[10] Ono, K. A stronger system of object theory as a prototype of set theory, Nagoya Math. J., vol. 22 (1963), pp. 119119.CrossRefGoogle Scholar
[11] Ono, K. On a practical way of describing formal deductions, Nagoya Math. J., vol. 21 (1962), pp. 115115.Google Scholar
[12] Suetuna, Z. Über die Grundlagen der Mathematik I-III: I, J. Math. Soc. Jap., vol. 3 (1951), pp. 5959: II, HI, Proc. Jap. Academy: II, vol. 27 (1951), pp. 389389: III, vol. 29 (1953), pp. 9191.Google Scholar