Natural number

The set of natural numbers, denoted $\mathbb{N}$ , is the set most conveniently associated with the notion of "counting".

Definition

Let $\mathcal{F}$ be the set of all successor sets $S$ .

The set of Natural Numbers $\mathbb{N}$ is defined as $\mathbb{N}=\bigcap_{S\in\mathcal{F}} S$

Note that as $1\in S$ $\forall S\in\mathcal{F}$ , $\mathbb{N}$ is non-empty.

Common Usage

According to this definition, $\mathbb{N}$ is the set $\{1,2,3,\ldots\}$ (Which is also called the set of counting numbers or positive integer)s. Unfortunately, in some texts, $\mathbb{N}$ is taken to be the set of whole numbers or nonnegative integers. Because of this ambiguity, one should always be careful to define one's notation clearly. Possible alternatives include $\mathbb{Z}_{\geq0}$ for the non-negative integers and $\mathbb{Z}_{>0}$ or $\mathbb{P}$ for the positive integers (although $\mathbb{P}$ is also sometimes used for the prime numbers). Natural numbers are important in the link between the well-ordering principle and the principle of mathematical induction.