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, $\left| {{{\cal S}_{{\rm{fin}}}}\left( x \right)} \right| < \left| {{\cal S}\left( x \right)} \right|$ 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, $\left| {{\rm{seq}}\left( x \right)} \right| < \left| {{\cal S}\left( x \right)} \right|$ 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, $|{[x]^2}| < \left| {{\cal S}\left( x \right)} \right|$.

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ZF 第一部分中无限红衣主教的因子：ZF 结果

对于一套X， 让${\cal S}\left(x\right)$是所有排列的集合X. 我们在 ZF（没有选择公理）中证明了关于这个概念的几个结果，其中包括：（1）对于所有集合X这样${\cal S}\left(x\right)$是戴德金无限的，$\左| {{{\cal S}_{{\rm{fin}}}}\left( x \right)} \right| < \左| {{\cal S}\left( x \right)} \right|$并且没有从${\cal S}\left(x\right)$进入${{\cal S}_{{\rm{fin}}}}\left( x \right)$， 在哪里${{\cal S}_{{\rm{fin}}}}\left( x \right)$表示所有排列的集合X它只移动有限多个元素。(2) 对于所有集合X这样${\cal S}\left(x\right)$是戴德金无限的，$\左| {{\rm{seq}}\left( x \right)} \right| < \左| {{\cal S}\left( x \right)} \right|$并且没有从${\cal S}\left(x\right)$进入序列（X), 其中 seq (X) 表示所有元素的有限序列的集合X.(3) 对于所有无限集X使得存在一个排列X没有不动点，就没有有限对一的函数${\cal S}\left(x\right)$进入X.(4) 对于所有集合X,$|{[x]^2}| < \左| {{\cal S}\left( x \right)} \right|$.