The finite sequences and the partitions whose members are finite of a set

2023/12/03 by Palagorn Phansamdaeng, Phansamdaeng, Palagorn, Pimpen Vejjajiva +1
Computer Science · Mathematics · #03E10 #Advanced Algebra and Logic #Advanced Topology and Set Theory #FOS: Mathematics #Logic (math.LO) #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.2312.01333

openalex publication_date 2023/12/03 · openalex created_date 2023/12/06 · openalex updated_date 2026/07/28

Abstract

In this paper, we investigate relationships between |\seq(A)| and |\Part_\fin(A)| in the absence of the Axiom of Choice, where \seq(A) is the set of finite sequences of elements in a set A and \Part_\fin(A) is the set of partitions of A whose members are finite. We show that |\seq(A)|<|\Part_\fin(A)| if A is Dedekind-infinite and the condition cannot be removed. Moreover, this relationship holds for an arbitrary infinite set A if we restrict \seq(A) to the set of finite sequences with a bounded length.

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