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I would like to specify the set $\{0, 1, 2, \dots\}$, i.e., non-negative integers in an engineering conference paper. Which symbol is more preferable?

  • $\mathbb{N}_0$
  • $\mathbb{N}\cup\{0\}$
  • $\mathbb{Z}_{\ge 0}$
  • $\mathbb{Z}_{+}$
  • $\mathbb{Z}_{0+}$
  • $\mathbb{Z}_{*}$
  • $\mathbb{Z}_{\geq}$
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    $\begingroup$ They all seem clear enough to me, except maybe $\mathbb{Z}_+$, which might not include $0$ :/ $\endgroup$ Mar 12, 2014 at 18:37
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    $\begingroup$ In my opinion, a notation using $\mathbb{Z}$ (such as $\mathbb{Z}_{\geq 0}$) is preferable over a notation using $\mathbb{N}$, a symbol that means different things in different countries. $\endgroup$
    – user133281
    Mar 12, 2014 at 18:39
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    $\begingroup$ $\mathbb{Z}_+$ looks like the set of strictly positive integers to me. $\mathbb{N}\cup \{0\}$ is unambiguous, even if it is redundant ('cause, you know, $0\in\mathbb{N}$). $\mathbb{Z}_{\geqslant 0}$ is also clear. $\endgroup$ Mar 12, 2014 at 18:39
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    $\begingroup$ @DanielFischer. Some people use the definition that $0\notin \mathbb{N}$. Hence, $\mathbb{N}$ alone is ambiguous. $\endgroup$ Aug 19, 2015 at 2:06
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    $\begingroup$ You forgot $\omega$! $\endgroup$
    – user29123
    Aug 19, 2015 at 2:35

6 Answers 6

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According to Wikipedia, unambiguous notations for the set of non-negative integers include $$ \mathbb{N}^0 = \mathbb{N}_0 = \{ 0, 1, 2, \ldots \}, $$ while the set of positive integers may be denoted unambiguously by $$ \mathbb{N}^* = \mathbb{N}^+ = \mathbb{N}_1 = \mathbb{N}_{>0}= \{ 1, 2, \ldots \}. $$

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Based on this similar post, the following seems to be preferred:

$\mathbb{Z}_{\geq 0}$

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Wolfram Mathworld has $\mathbb{Z}^*$.

Nonnegative integer

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    $\begingroup$ I might interpret that as either the nonzero integers or as the group of units of the integers. $\endgroup$ Oct 19, 2015 at 22:25
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    $\begingroup$ Right, those are the things I would interpret $\mathbb{Z}^{\ast}$ as. Very confusing notation on Mathworld. $\endgroup$ Oct 19, 2015 at 22:26
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    $\begingroup$ In my opinion, it's a bad notation; but this answer is valid, since it references Wolfram Mathworld, which is a popular and reliable source. $\endgroup$
    – Wood
    Aug 7, 2016 at 4:55
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I personally always use $\Bbb N_0$ because what you are really describing is just the natural numbers plus the element $\{0\}$.

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In set theory, the natural numbers are understood to include $0$. The set of natural numbers $\{0,1,2,\dots\}$ is often denoted by $\omega$.

There are two caveats about this notation:

  • It is not commonly used outside of set theory, and it might not be recognised by non-set-theorists.
  • In "everyday mathematics", the symbol $\mathbb N$ is rarely used to refer to a specific model of the natural numbers. By contrast, $\omega$ denotes the set of finite von Neumann ordinals: $0=\varnothing$, $1=\{0\}$, $2=\{0,1\}$, $3=\{0,1,2\}$, etc. This is a specific construction of the natural numbers in which they are defined as certain sets.
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Many authors consider $0$ to be a natural number, and accordingly use $\mathbb N$ to denote the set of nonnegative integers. This is especially common in mathematical logic, set theory, combinatorics and some branches of algebra (but not so common in analysis or applied mathematics). Usage also depends on the country: I find that in Europe, $0$ is more likely to be included in the naturals than it is in the US.

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  • $\begingroup$ Asia also does not include $0$ in $\mathbb{N}$. $\endgroup$
    – IraeVid
    Dec 1, 2023 at 3:55

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