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$\mathbb Z^+$ stands for the Positive Integers: $\{1,2,3,4,5\dots\}$

$\mathbb N$ stands for the Natural Numbers: $\{1,2,3,4,5\dots\}$

So what is the difference between $\mathbb Z^+$ and $\mathbb N$?

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Many people (myself included) would use $\mathbb{N} = \{0, 1, 2, 3, \dots\}$. –  T. Bongers Nov 17 '13 at 7:27
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Many people (myself included) would consider Bongers weird to do that. But he is right. The difference would otherwise be, in my mind at least, that $\Bbb Z^+$ supposes that there is the ambient space $\Bbb Z$ that it sits inside of, whereas $\Bbb N$ stands on its own (by default). It's like saying "vector subspace" instead of just "vector space". A subspace is a vector space in its own right, but by specifically invoking the "sub" you are indicating that there is a larger object of potential relevance that this thing is inside of. Practically speaking, no difference, though. –  zibadawa timmy Nov 17 '13 at 7:32
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@zibadawa: Many people would like to say that a set is finite if the number of elements it has is a natural number. If zero is not a natural number, the empty set is not finite. What is it, then? –  Asaf Karagila Nov 17 '13 at 7:37
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@zibadawatimmy I personally like my monoids to have unit elements, so I vote for $\mathbb N$ to include $0$. What is your reason to find that weird? –  Ittay Weiss Nov 17 '13 at 7:57
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I'm assuming that when the OP says "difference," he or she is not referring to set difference. :P –  Joel Reyes Noche Nov 17 '13 at 8:56
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According to Wikipedia, the natural numbers $\mathbb{N}$ are sometimes thought of as the positive integers $\mathbb{Z}^+=\{1,2,3,\dots\}$ or as the non-negative integers $\{0,1,2,\dots\}$. That is why mathematicians should always clearly define what they mean by natural numbers at the start.

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