---
tag: 0MWKJ44V5A
kind: object
name: Natural numbers
aliases: [natural-number, set-of-natural-numbers]
msc: [11A99]
sources:
  - url: https://en.wikipedia.org/wiki/Natural_number
    locator: lead section
  - url: https://en.wikipedia.org/wiki/Natural_number
    locator: Formal definitions
same_as:
  wikidata: Q21199
  wikipedia: Natural number
---

## Description

The natural numbers are the numbers $0, 1, 2, 3, \dots$ that count the elements of a finite set.
We write $\mathbb{N} = \{0, 1, 2, 3, \dots\}$ for the set of natural numbers.
Each natural number $n$ has a next natural number $n + 1$, its successor.
The [[Peano axioms|0QFT7Z76N5]] describe $\mathbb{N}$ with the number $0$ and the [[successor function|0AK9TGSNXY]].
The [[addition|0S13V32CPB]], the [[multiplication|06J5DE801P]] and the [[order|0HNATXFEV5]] of $\mathbb{N}$ come from $0$ and the successor function.

## Remarks

Some authors start the natural numbers at $1$. They then write $\mathbb{N}_0$ or $\mathbb{Z}_{\ge 0}$ for the set $\{0, 1, 2, \dots\}$.
Every nonempty set of natural numbers has a least element. This is the [[well-ordering principle|01BX15HSRR]].
The [[integers|0C4JS9RF5M]] contain the natural numbers as the integers that are not negative.

## Explanations

### L1 human

The natural numbers are the numbers that you use to count: zero, one, two, three, and so on.
There is no largest natural number, because you can always add one more.
All the other number systems of this course, such as the integers and the real numbers, start from the natural numbers.