Answer: In my experience: the matter of 0∈ℕ0∈N depends on what you want to do with the set ℕN. If you want to use its algebraic properties, especially number theory, then usually 0∉ℕ0∉N is nicer. There is no particular reason to include it, and it will often create the need for special cases in theorems. Example: the fundamental theorem of arithmetic. Example: the equivalence between irreducibility and primality.
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