Symbols:N

Set of Natural Numbers

 * $\N$

The set of natural numbers:
 * $\N = \set {0, 1, 2, 3, \ldots}$

Set of Non-Zero Natural Numbers

 * $\N_{> 0}$ or $\N_{\ne 0}$

The set of non-zero natural numbers:
 * $\N_{> 0} = \set {1, 2, 3, \ldots}$

Deprecated

 * $\N^*$

The set of non-zero natural numbers:
 * $\N^* = \set {1, 2, 3, \ldots}$.

Initial Segment of Natural Numbers

 * $\N_n$, $\N^*_n$

The set $\N_n$ is the set of all natural numbers which are less than $n$:
 * $\N_n = \set {x \in \N: x < n} = \set {0, 1, 2, \ldots, n - 1}$

Similarly, the set $\N^*_n$ is the set of all non-zero natural numbers which are less or equal to $n$:
 * $\N^*_n = \set {x \in \N^*: x \le n} = \set {1, 2, \ldots, n}$

Deprecated Usages
Older literature tends to use $\N$ to mean $\set {1, 2, 3, \ldots}$.

Consequently, the set $\set {0, 1, 2, 3, \ldots}$ needs another symbol to denote it. The usual technique is to use $\tilde \N$.