Do the natural numbers include zero? And what should we call them, with or without zero?

Asked Dec 12 '10 at 15:52

Active May 23 '11 at 16:23

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Is 0 a natural number?

There seems to be no consensus, although perhaps one is gathering over the centuries to say yes to the first question and identify $\mathbb{N}$ with $\omega$ : the set of finite Von Neumann ordinal numbers. That leaves $\textbf{N}$ = {$1$, $2$, ...} for the counting numbers (who counts with $0$?). You can dodge the issue by writing $\mathbb{Z}_{\geq0}$, $\mathbb{Z}_+$, etc., but it seems awkward to use derived and complicated notation for something so basic.

edited Apr 13 '17 at 12:21

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asked Dec 12 '10 at 15:52

John Bentin

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8

Oh boy... – J. M. ain't a mathematician Dec 12 '10 at 15:54
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"who counts with 0?" - C/++ programmers? – J. M. ain't a mathematician Dec 12 '10 at 15:56
@J.M:Probably you meant C/C++ but actually there is no such language,contiguous array indexing start from $0$ not only in C and C++ but also in D,Java and probably in few other languages. – Quixotic Dec 12 '10 at 16:06
@Deb: It's a tic of mine; I'm too lazy to write another C. :) In any event, strings are also indexed at 0 (okay fine, they're arrays too :P ). – J. M. ain't a mathematician Dec 12 '10 at 16:08
@J.M: Cool! ;-) – Quixotic Dec 12 '10 at 16:09
@J.M.: Well, it sounds strange to call "0" the "first" number. – Hendrik Vogt Dec 12 '10 at 16:13
I agree @Hendrik, I'm just repeating an old lame joke... :) – J. M. ain't a mathematician Dec 12 '10 at 16:14
@J.M:In C strings are actually null terminated character arrays but yes they are also indexed from $0$. – Quixotic Dec 12 '10 at 16:17
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The only programming language known to me that does not count from 0 is the one used in Matlab. In fact, indexing arrays from 1 makes little to no sense. In CS, typically $\mathbb{N} = {0, 1, \dots}$ is used. – Raphael Dec 12 '10 at 18:07

Do the natural numbers include zero? And what should we call them, with or without zero?

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