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0 simply means some object that when combined with an appropriate successor function satisfies the Peano axioms. Intuitively S a is a 1. S is injective i.e. distinct natural numbers have distinct successors if a b then S a S b. Under this definition there are exactly n elements in the na ve sense in the set n and n m in the na ve sense if and only if n is a subset of m. Also with this definition different possible interpretations of notations like Rn n-tuples versus mappings of n into R...

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