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