For a set x, let ${\cal S}\left( x \right)$ be the set of all permutations of x. We prove by the method of permutation models that the following statements are consistent with ZF:(1) There is an infinite set x such that $|\wp \left( x \right)| < |{\cal S}\left( x \right)| < |se{q^{1 - 1}}\left( x \right)| < |seq\left( x \right)|$, where $\wp \left( x \right)$ is the power set of x, seq (x) is the set of all finite sequences of elements of x, and seq1-1 (x) is the set of all finite sequences of elements of x without repetition.(2) There is a Dedekind infinite set x such that $|{\cal S}\left( x \right)| < |{[x]^3}|$ and such that there exists a surjection from x onto ${\cal S}\left( x \right)$.(3) There is an infinite set x such that there is a finite-to-one function from ${\cal S}\left( x \right)$ into x.

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ZF 第 II 部分中无限红衣主教的因子：一致性结果

对于一套X， 让${\cal S}\left(x\right)$是所有排列的集合X. 我们通过置换模型的方法证明了下列陈述与 ZF 一致：(1) 存在一个无限集X这样$|\wp\left(x\right)| < |{\cal S}\left( x \right)| < |se{q^{1 - 1}}\left( x \right)| < |seq\left( x \right)|$， 在哪里$\wp\left(x\right)$是幂集X, seq (x) 是元素的所有有限序列的集合X, 和序列1-1(x) 是所有元素的有限序列的集合X(2) 有一个 Dedekind 无限集X这样$|{\cal S}\left( x \right)| < |{[x]^3}|$并且存在一个从X到${\cal S}\left(x\right)$.(3) 有一个无限集X使得有一个有限对一的函数${\cal S}\left(x\right)$进入X.