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

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.