# Mathematical Logic Quarterly

## Relations between cardinalities of the finite sequences and the finite subsets of a set

### Abstract

We write $\text{seq}\left(\mathfrak{m}\right)$ and $\text{fin}\left(\mathfrak{m}\right)$ for the cardinalities of the set of finite sequences and the set of finite subsets, respectively, of a set which is of cardinality $\mathfrak{m}$. With the axiom of choice ( $\mathsf{AC}$), $\text{seq}\left(\mathfrak{m}\right)=\text{fin}\left(\mathfrak{m}\right)$ for every infinite cardinal $\mathfrak{m}$ but, without $\mathsf{AC}$, any relationship between $\text{seq}\left(\mathfrak{m}\right)$ and $\text{fin}\left(\mathfrak{m}\right)$ for an arbitrary infinite cardinal $\mathfrak{m}$ cannot be proved. In this paper, we give conditions that make $\text{seq}\left(\mathfrak{m}\right)$ and $\text{fin}\left(\mathfrak{m}\right)$ comparable for an infinite cardinal $\mathfrak{m}$. Among our results, we show that, if we assume the axiom of choice for sets of finite sets, then $\text{seq}\left(\mathfrak{m}\right)=\text{fin}\left(\mathfrak{m}\right)$ for every Dedekind‐infinite cardinal $\mathfrak{m}$ and the condition that $\mathfrak{m}$ is Dedekind‐infinite cannot be weakened to weakly Dedekind‐infinite.

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