 Dedekind’s definition is only one among several alternative definitions of infinite set (and correspondingly of finite set) that have been proposed by him and after him. If one assumes the Axiom of Choice, these alternative definitions turn out to be equivalent. (This axiom says that for every set \(A\) of pairwise-disjoint non-empty sets, there exists a function that selects exactly one element from each set in \(A\).) But without the Axiom of Choice, the definitions can be shown not to be equivalent and the foundational situation is rather delicate but well understood (see Moore 1982).