Definition:Natural Numbers

Informal Definition
The natural numbers are the counting numbers.

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

The set $\N \setminus \left\{{0}\right\}$ is denoted $\N_{>0}$:
 * $\N_{>0} = \left\{{1, 2, 3, \ldots}\right\}$

The set of natural numbers is one of the most important sets in mathematics.

1-Based Natural Numbers
The following axioms are intended to capture the behaviour of $\N_{>0}$, the element $1 \in \N_{>0}$, and the operations $+$ and $\times$ as they pertain to $\N_{>0}$:

Also known as
First, note that some sources use a different style of letter from $\N$: you will find $N$, $\mathbf N$, etc. However, $\N$ is becoming more commonplace and universal nowadays.

The usual symbol for denoting $\left\{{1, 2, 3, \ldots}\right\}$ is $\N^*$, but the more explicit $\N_{>0}$ is standard on.

Some authors refer to $\left\{{0, 1, 2, 3, \ldots}\right\}$ as $\tilde {\N}$, and refer to $\left\{{1, 2, 3, \ldots}\right\}$ as $\N$.

Either is valid, and as long as it is clear which is which, it does not matter which is used. However, using $\N = \left\{{0, 1, 2, 3, \ldots}\right\}$ is a more modern approach, particularly in the field of computer science, where starting the count at zero is usual.

Treatments which consider the natural numbers as $\left\{{1, 2, 3, \ldots}\right\}$ sometimes refer to $\left\{{0, 1, 2, 3, \ldots}\right\}$ as the positive (or non-negative) integers $\Z_{\ge 0}$.

The following notations are sometimes used:
 * $\N_0 = \left\{{0, 1, 2, 3, \ldots}\right\}$
 * $\N_1 = \left\{{1, 2, 3, \ldots}\right\}$

However, beware of confusing this notation with the use of $\N_n$ as the initial segment of the natural numbers:
 * $\N_n = \left\{{0, 1, 2, \ldots, n-1}\right\}$

under which notational convention $\N_0 = \varnothing$ and $\N_1 = \left\{{0}\right\}$.

So it is important to ensure that it is understood exactly which convention is being used.

The use of $\N$ or its variants is not universal, either.

Some sources, for example uses $P = \left\{{1, 2, 3, \ldots}\right\}$.

This may stem from the fact that Jacobson's presentation starts with Peano's axioms.

On the other hand, it may just be because $P$ is the first letter of positive.

Based on defining $\N$ as being the minimal infinite successor set $\omega$, suggests using $\omega$ for the set of natural numbers.

This use of $\omega$ is usually seen for the order type of the natural numbers, that is, $\left({\N, \le}\right)$ where $\le$ is the usual ordering on the natural numbers.

Also see

 * Definition:Natural Number Addition