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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?

  • $\mathbb{N}_0$
  • $\mathbb{N}\cup\{0\}$
  • $\mathbb{Z}_{\ge 0}$
  • $\mathbb{Z}_{+}$
  • $\mathbb{Z}_{0+}$
  • $\mathbb{Z}_{*}$
  • $\mathbb{Z}_{\geq}$
InanimateBeing
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Ari
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    They all seem clear enough to me, except maybe $\mathbb{Z}_+$, which might not include $0$ :/ – G Tony Jacobs Mar 12 '14 at 18:37
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    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. – user133281 Mar 12 '14 at 18:39
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    $\mathbb{Z}_+$ 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. – Daniel Fischer Mar 12 '14 at 18:39
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    @DanielFischer. Some people use the definition that $0\notin \mathbb{N}$. Hence, $\mathbb{N}$ alone is ambiguous. – Batominovski Aug 19 '15 at 02:06
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    You forgot [$\omega$](https://en.wikipedia.org/wiki/Ordinal_number)! –  Aug 19 '15 at 02:35
  • For me is it just ℕ as it is the common use in France – Lelouch Oct 08 '22 at 13:53

5 Answers5

24

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 $$ \mathbb{N}^* = \mathbb{N}^+ = \mathbb{N}_1 = \mathbb{N}_{>0}= \{ 1, 2, \ldots \}. $$

A. Donda
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14

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

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

Community
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Pablo Rivas
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2

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

Nonnegative integer

David G
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    I might interpret that as either the nonzero integers or as the group of units of the integers. – Qiaochu Yuan Oct 19 '15 at 22:25
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    Right, those are the things I would interpret $\mathbb{Z}^{\ast}$ as. Very confusing notation on Mathworld. – Daniel Fischer Oct 19 '15 at 22:26
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    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. – Wood Aug 07 '16 at 04:55
1

I personally always use $\Bbb N_0$ because what you are really describing is just the natural numbers plus the element $\{0\}$.

Aaron Hendrickson
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1

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.
Joe
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