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An Axiomatic Theory of Ordinal Numbers

Published online by Cambridge University Press:  22 January 2016

Toshio Umezawa
Toshio Umezawa*
Affiliation:
Mathematical Institute, Nagoya University
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In this paper, a formal theory of ordinal numbers is developed on an axiomatic basis whose details are described in § 1. Our primitive notions are set, class, collection, a binary relation ∈, and collection formation {!}. Sets and classes in our theory play similar roles as sets and classes respectively in Gödel [1] except the difference that an element of a class is a class but not necessarily a set. A new notion, introduced into our theory is that of collections. A collection relates to a class, just as a class relates to a set in von Neumann’s theory. That is; a set is a class and a class is a collection but the converses are not generally the case. For example, all the natural numbers, all the real numbers etc. constitute sets, the ordinal numbers which are sets constitute a proper class, and the totality of ordinal numbers as well as that of all classes are proper collections. These relations are described by axiom group (A).

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 18 , April 1961 , pp. 193 - 228
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1961

References

[1] Gödel, K., The consistency of the axiom of choice and of the generalized continuum-hypothesis with the axioms of set theory, Princeton, 1940.Google Scholar
[2] Kuroda, S., An investigation on the logical structure of mathematics. Part (I), Hamb. Abh. 22 (1958), pp. 249–266.Google Scholar
[3] Quine, W. V., Mathematical logic, New York, 1940.Google Scholar
[4] Takeuti, G., On a generalized logic calculus, Japanese J. of Math., 23 (1953), pp. 39–96.CrossRefGoogle Scholar
[5] Umezawa, T., On logics intermediate between intuitionistic and classical predicate logic, J. Symbolic Logic, 24(1959), pp. 141–153.Google Scholar
[6] Wang, H., On Zermelo’s and von Neumann’s axioms for set theory, Proc. Nat. Acad. Sci. U.S.A., 35(1949), pp. 150–155.CrossRefGoogle ScholarPubMed
[7] Wang, H., Between number theory and set theory, Math. Ann., 126 (1953), pp. 885–409.Google Scholar