FACTORIALS OF INFINITE CARDINALS IN ZF PART I: ZF RESULTS.

For a set x , let ${\cal S}\left(x \right)$ be the set of all permutations of x. We prove in ZF (without the axiom of choice) several results concerning this notion, among which are the following: (1) For all sets x such that ${\cal S}\left(x \right)$ is Dedekind infinite, $\left| {{{\cal S}_{{\rm{fin}}}}\left(x \right)} \right| and there are no finite-to-one functions from ${\cal S}\left(x \right)$ into ${{\cal S}_{{\rm{fin}}}}\left(x \right)$ , where ${{\cal S}_{{\rm{fin}}}}\left(x \right)$ denotes the set of all permutations of x which move only finitely many elements. (2) For all sets x such that ${\cal S}\left(x \right)$ is Dedekind infinite, $\left| {{\rm{seq}}\left(x \right)} \right| and there are no finite-to-one functions from ${\cal S}\left(x \right)$ into seq (x), where seq (x) denotes the set of all finite sequences of elements of x. (3) For all infinite sets x such that there exists a permutation of x without fixed points, there are no finite-to-one functions from ${\cal S}\left(x \right)$ into x. (4) For all sets x , $|{[x]^2}|. [ABSTRACT FROM AUTHOR]