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

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