Hostname: page-component-6bf8c574d5-mggfc Total loading time: 0 Render date: 2025-02-17T18:28:06.252Z Has data issue: false hasContentIssue false
  • English
  • Français

A FINITE-TO-ONE MAP FROM THE PERMUTATIONS ON A SET

Part of: Set theory

Published online by Cambridge University Press:  19 October 2016

NATTAPON SONPANOW  and
PIMPEN VEJJAJIVA
NATTAPON SONPANOW
Affiliation:
Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University, Bangkok, Thailand email [email protected]
PIMPEN VEJJAJIVA*
Affiliation:
Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University, Bangkok, Thailand email [email protected]
*
✉[email protected]
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.

Forster [‘Finite-to-one maps’, J. Symbolic Logic68 (2003), 1251–1253] showed, in Zermelo–Fraenkel set theory, that if there is a finite-to-one map from ${\mathcal{P}}(A)$, the set of all subsets of a set $A$, onto $A$, then $A$ must be finite. If we assume the axiom of choice (AC), the cardinalities of ${\mathcal{P}}(A)$ and the set $S(A)$ of permutations on $A$ are equal for any infinite set $A$. In the absence of AC, we cannot make any conclusion about the relationship between the two cardinalities for an arbitrary infinite set. In this paper, we give a condition that makes Forster’s theorem, with ${\mathcal{P}}(A)$ replaced by $S(A)$, provable without AC.

Keywords

finite-to-one mappermutationaxiom of choice

MSC classification

Primary: 03E10: Ordinal and cardinal numbers
Secondary: 03E25: Axiom of choice and related propositions
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 95 , Issue 2 , April 2017 , pp. 177 - 182
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

References

Dawson, J. W. Jr. and Howard, P. E., ‘Factorials of infinite cardinals’, Fund. Math. 93 (1976), 186–195.CrossRefGoogle Scholar
Enderton, H. B., Elements of Set Theory (Academic Press, New York, 1977).Google Scholar
Forster, T., ‘Finite-to-one maps’, J. Symbolic Logic 68 (2003), 1251–1253.CrossRefGoogle Scholar
Jech, T. J., The Axiom of Choice (North-Holland, Amsterdam, 1973).Google Scholar
Sageev, G., ‘An independence result concerning the axiom of choice’, Ann. Math. Logic 8 (1975), 1–184.CrossRefGoogle Scholar