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Definition: A set S is finite if it is empty or there is a bijection to Nn for some n > 0. We then call 0 resp. n the size or cardinality of S:
A set is infinite if is not finite:
S is finite : <=> S = 0 \/ (exists n in N>0, f: f: Nn ->bijective S); |S| := if S = 0 then 0 else (such n in N>0: exists f: f: Nn ->bijective S).
S is infinite : <=> ~S is finite.