Alternative Definition of an Infinite Set

A set A is infinite if and only if it is equivalent to one of its proper subsets

Explanation

This is the second definition of an infinite set (a positive characterisation): a set \(A\) is infinite if and only if \(A \sim B\) for some proper subset \(B \subsetneq A\). Example: \(\mathbb{N} \sim \{2,4,6,\ldots\}\) via \(f(n)=2n\), and \(\{2,4,6,\ldots\}\) is a proper subset of \(\mathbb{N}\). No finite set can do this.