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the funny thing about 0 is that in set theory it's used to construct all of the other positive integers. you start with 0 defined as the empty set. then 1 is {0}; that is, it's the set whose only element is 0. then 2 is {0, {0}} etc. these are called the "von neumann ordinals"
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for a pure mathematician 0, 1, and infinity have the special feature that they're the only points that are always defined on the projective line P^1(k) over an arbitrary field k (they're the only points when k = F_2) although that's a bit abstruse
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