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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]
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Abstract

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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

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