Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-27T15:55:21.061Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

[1]Hickman, J.L., “On the set of finite subsets of a set”, submitted.Google Scholar
[2]Rubin, Herman and Rubin, Jean E., Equivalents of the axiom of choice (North-Holland, Amsterdam, 1963).Google Scholar
[3]Shoenfield, J.R., “Unramified forcing”, Axiomatic set theory, 357–381 (Proc. Sympos. Pure Math., Vol. 13, Part 1, Univ. California, Los Angeles, Calif., 1967; Amer. Math. Soc., Providence, Rhode Island, 1971).CrossRefGoogle Scholar
[4]Tarski, Alfred, “Sur les ensembles finis”, Fund. Math. 6 (1924), 45–95.CrossRefGoogle Scholar