Mathematical Logic Quarterly

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

Journal Article

Abstract

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

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