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A Stronger System of Object Theory as a Prototype of Set Theory

Published online by Cambridge University Press:  22 January 2016

Katuzi Ono
Katuzi Ono*
Affiliation:
Mathematical Institute, Nagoya University
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We have introduced in our former work [1] a theory of mathematical objects which can be regarded as a prototype of set theory. We have been successful to imbed the Zermelo set-theory [3] without the axiom of choice in our system. However, it seems impossible to imbed the Fraenkel set-theory [4] in our system even without the axiom of choice. In this work, we introduce another system of object theory in which we can imbed the Fraenkel set-theory without the axiom of choice. We shall denote our former system by OZ (object theory in the manner of the Zermelo set-theory) and the new system we are going to introduce in this work by OF (object theory in the manner of the Fraenkel set-theory). We shall also denote the Zermelo set-theory without the axiom of choice by SZ and the Fraenkel set-theory without the axiom of choice by SF.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 22 , June 1963 , pp. 119 - 167
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1963

References

[1] Ono, K., A theory of mathematical objects as a prototype of set theory, Nagoya Math. Jour., vol. 20 (1962), pp. 105–168.Google Scholar
[2] Ono, K., On a practical way of describing formal deductions, Nagoya Math. Jour., vol. 21 (1962), pp. 115–121.Google Scholar
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[4] Fraenkel, A., Einlitung in die Mengenlehre, 1928 (3. Aufl.)Google Scholar
[5] Godei, 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
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