The partitions whose members are finite and the permutations with at most n non-fixed points of a set

2023/12/03 by Sonpanow, Nattapon, Vejjajiva, Pimpen
#03E10 #FOS: Mathematics #Logic (math.LO)

paper · doi:10.48550/arxiv.2312.01349

Abstract

We write S_{≤ n}(A) and \Part_\fin(A) for the set of permutations with at most n non-fixed points, where n is a natural number, and the set of partitions whose members are finite, respectively, of a set A. Among our results, we show, in the Zermelo-Fraenkel set theory, that |\Part_\fin(A)| \nleq |S_{≤ n}(A)| for any infinite set A and if A can be linearly ordered, then |S_{≤ n}(A)| < |\Part_\fin(A)| while the statement ``|S_{≤ n}(A)|≤|\Part_\fin(A)| for all infinite sets A" is not provable for n≥ 3.

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