Hostname: page-component-745bb68f8f-f46jp Total loading time: 0 Render date: 2025-01-12T23:06:21.480Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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