Definition:Natural Numbers/Notation

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Informal Definition

The notation for the set of natural numbers varies in the literature.

Some sources use a different style of letter from $\N$: you will find $N$, $\mathbf N$, etc. However, $\N$ is commonplace and all but universal.

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

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

Either is valid, and as long as it is clear which is which, it does not matter which is used. However, using $\N = \set {0, 1, 2, 3, \ldots}$ 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 $\set {1, 2, 3, \ldots}$ sometimes refer to $\set {0, 1, 2, 3, \ldots}$ as the positive (or non-negative) integers $\Z_{\ge 0}$.

The following notations are sometimes used:

$\N_0 = \set {0, 1, 2, 3, \ldots}$
$\N_1 = \set {1, 2, 3, \ldots}$

However, beware of confusing this notation with the use of $\N_n$ as the initial segment of the natural numbers:

$\N_n = \set {0, 1, 2, \ldots, n - 1}$

under which notational convention $\N_0 = \O$ and $\N_1 = \set 0$.

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 Nathan Jacobson: Lectures in Abstract Algebra: Volume $\text { I }$: Basic Concepts ($1951$) uses $P = \set {1, 2, 3, \ldots}$.

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.

Similarly, Undergraduate Topology by Robert H. Kasriel uses $\mathbf P$, which he describes as the set of all positive integers.

Based on defining $\N$ as being the minimally inductive set $\omega$, 1960: Paul R. Halmos: Naive Set Theory suggests using $\omega$ for the set of natural numbers.

This convention is followed by Raymond M. Smullyan and Melvin Fitting in their Set Theory and the Continuum Problem.

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