The Free On-line Dictionary of Computing (30 December 2018):

infinite set

    A set with an infinite number of elements.
   There are several possible definitions, e.g.

   (i) ("Dedekind infinite") A set X is infinite if there exists
   a bijection (one-to-one mapping) between X and some proper
   subset of X.

   (ii) A set X is infinite if there exists an injection from N
   (the set of natural numbers) to X.

   In the presence of the Axiom of Choice all such definitions
   are equivalent.

   (1995-03-27)