The power set and the set of permutations with finitely many non-fixed points of a set

Pub Date : 2023-05-10 DOI:10.1002/malq.202100070

Guozhen Shen

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引用次数: 3

Abstract

For a cardinal $a$ , we write $S_{fin} (a)$ for the cardinality of the set of permutations with finitely many non-fixed points of a set which is of cardinality $a$ . We investigate the relationships between $2^{a}$ and $S_{fin} (a)$ for an arbitrary infinite cardinal $a$ in $ZF$ (without the axiom of choice). It is proved in $ZF$ that $2^{a} \neq S_{fin} (a)$ for all infinite cardinals $a$ , and we show that this is the best possible result.

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幂集和集的具有有限多个不动点的置换集

对于基数$\mathfrak｛a｝$，我们写S fin（a）$\运算符名称｛\mathcal{S}_｛\text｛fin｝｝（\mathfrak｛a｝）$为基数为a$\mathfrak{a｝$的集合的具有有限多个非不动点的置换集的基数。我们研究了2a$2^\mathfrak｛a｝$与S fin之间的关系（a）$\运算符名称｛\mathcal{S}_｛\text｛fin｝｝（\mathfrak｛a｝）$，用于ZF$\mathsf｛ZF｝$中的任意无限基数a$\mathfrak{a｝$（没有选择公理）。在ZF$\mathsf{ZF}$中证明了2a≠S fin（a）$2^\mathfrak｛a｝\ ne \ operator name｛\mathcal{S}_｛\text｛fin｝｝（\mathfrak｛a｝）$对于所有无限基数a$\mathfrak{a｝$，我们证明这是最好的可能结果。

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