The best symbol for non-negative integers?

Question

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?

• $\mathbb{N}_0$
• $\mathbb{N}\cup\{0\}$
• $\mathbb{Z}_{\ge 0}$
• $\mathbb{Z}_{+}$
• $\mathbb{Z}_{0+}$
• $\mathbb{Z}_{*}$
• $\mathbb{Z}_{\geq}$

Answer 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.

Answer 2

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.

Answer 3

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

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

Answer 4

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

Answer 5

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

Nonnegative integer

Answer 6

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