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

Abstract

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.