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General-well-ordered sets

Published online by Cambridge University Press:  09 April 2009

J. L. Hickman
J. L. Hickman
Affiliation:
Department of Mathematics, Institute of Advanced Studies, Australian National University, Canberra, ACT
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It is of course well known that within the framework of any reasonable set theory whose axioms include that of choice, we can characterize well-orderings in two different ways: (1) a total order for which every nonempty subset has a minimal element; (2) a total order in which there are no infinite descending chains.

Now the theory of well-ordered sets and their ordinals that is expounded in various texts takes as its definition characterization (1) above; in this paper we commence an investigation into the corresponding theory that takes characterization (2) as its starting point. Naturally if we are to obtain any differences at all, we must exclude the axiom of choice from our set theory. Thus we state right at the outset that we are working in Zermelo-Fraenkel set theory without choice.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 19 , Issue 1 , February 1975 , pp. 7 - 20
Copyright
Copyright © Australian Mathematical Society 1975

References

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