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Small sets with large power sets

Published online by Cambridge University Press:  17 April 2009

G.P. Monro
G.P. Monro
Affiliation:
Department of Pure Mathematics, University of Sydney, Sydney, New South Wales.
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Abstract

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One problem in set theory without the axiom of choice is to find a reasonable way of estimating the size of a non-well-orderable set; in this paper we present evidence which suggests that this may be very hard. Given an arbitrary fixed aleph κ we construct a model of set theory which contains a set X such that if Y ⊆ X then either Y or X - Y is finite, but such that κ can be mapped into S(S(S(X))). So in one sense X is large and in another X is one of the smallest possible infinite sets. (Here S(X) is the power set of X.)

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 8 , Issue 3 , June 1973 , pp. 413 - 421
Copyright
Copyright © Australian Mathematical Society 1973

References

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