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Whether zero is a natural number or not is still controversial. What is the (mathematical) argument for 0 *NOT* being a natural number?

Mathematics
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Have you read http://math.stackexchange.com/questions/283/is-0-a-natural-number already?
My argument for \(0 \notin \mathbb{N}\) is simply because I find it more handy to deal with the notations this way, take for example the notation of a limit of a sequence \(a_n\): \[\Large \forall \epsilon > 0, \exists N \in \mathbb{N} \text{ such that } \forall n \geq N, |a_n-a| < \epsilon \] If you would define \(\mathbb{N}={0,1,2, \dots } \) then the above notation would require a slight update because \(N\) should be a positive number, whereas \(0\) accounts for a neutral number.
"In mathematics, there are two conventions for the set of natural numbers: it is either the set of positive integers {1,2,3,…} according to the traditional definition;" What is that traditional definition?

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Other answers:

positive integers is the traditional definition, 1,2,3,4,5,... also known as counting numbers, because it is how we traditionally count. While 0,1,2,3,4... are considered as integers including 0 which apparently came up in the 19th century.
How do you define "positive integers"?
positive integers \( \lbrace n \in \mathbb{Z} \mid n > 0 \rbrace= 1,2,3,4, \dots \)
where \(\mathbb{Z}=\lbrace...,-3,-2,-1,0,1,2,3,\dots \rbrace \)
Alright, thanks a lot for your help!

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