当前位置: X-MOL 学术J. Symb. Log. › 论文详情
Our official English website, www.x-mol.net, welcomes your feedback! (Note: you will need to create a separate account there.)
FACTORIALS OF INFINITE CARDINALS IN ZF PART I: ZF RESULTS
The Journal of Symbolic Logic ( IF 0.5 ) Pub Date : 2019-11-04 , DOI: 10.1017/jsl.2019.74
GUOZHEN SHEN , JIACHEN YUAN

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

中文翻译:

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,$|{[x]^2}| < \左| {{\cal S}\left( x \right)} \right|$.
更新日期:2019-11-04
down
wechat
bug