Dedekind-infinite set

A set \( S \) is called *Dedekind-infinite* if there exists a subset \( T \subseteq S \) such that there is a bijection between \( T \) and \( S \) itself (i.e., \( T \) can be put into one-to-one correspondence with \( S \)), and \( T \) is a proper subset of \( S \) (meaning \( T \) does not include all elements of \( S \)).