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

Abstract

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