Many sets that arise in mathematics do not allow a bijection to be established with {1, 2, ..., "n"} for "any" natural number "n"; these are called infinite sets, while those sets for which such a bijection does exist (for some "n") are called finite sets. Infinite sets cannot be counted in the usual sense; for one thing, the mathematical theorems which underlie this usual sense for finite sets are false for infinite sets. Furthermore, different definitions of the concepts in terms of which these theorems are stated, while equivalent for finite sets, are inequivalent in the context of infinite sets.