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

Forster [‘Finite-to-one maps’, J. Symbolic Logic 68 (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 cardinali...

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Published in:Bulletin of the Australian Mathematical Society 2017-04, Vol.95 (2), p.177-182
Main Authors: SONPANOW, NATTAPON, VEJJAJIVA, PIMPEN
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Set theory
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description Forster [‘Finite-to-one maps’, J. Symbolic Logic 68 (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.
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title A FINITE-TO-ONE MAP FROM THE PERMUTATIONS ON A SET
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