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Short definitions of the ordinals1

Published online by Cambridge University Press:  12 March 2014

Kenneth R. Brown  and
Hao Wang
Kenneth R. Brown
Affiliation:
Harvard University
Hao Wang
Affiliation:
Harvard University
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Extract

In this paper, a simple inductive characterization of the ordinal numbers is stated and developed. The characterization forms the basis for a set of axioms for ordinal theory and also for several short explicit definitions of the ordinals. The axioms are shown to be sufficient for ordinal theory, and, subject to suitable existence assumptions, each of the definitions is shown to imply the axioms.

The present results apply to the familiar von Neumann version of the ordinals, but the methods used are easily adapted to other versions.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 31 , Issue 3 , 02 September 1966 , pp. 409 - 414
Copyright
Copyright © Association for Symbolic Logic 1996

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Footnotes

1

Work for this paper was supported in part by the National Science Foundation, Grant GP-228.

References

[1]Bernays, P. and Fraenkel, A. A., Axiomatic set theory, Amsterdam (1958).Google Scholar
[2]Quine, W. V. and Wang, Hao, On ordinals, Bulletin of the American Mathematical Society, vol. 70 (1964), pp. 297f.CrossRefGoogle Scholar
[3]Quine, W. V., Set theory and its logic, Cambridge, Mass., Harvard University Press (1963).CrossRefGoogle Scholar