---
tag: 0Y9H52BT03
kind: definition
name: Order on the natural numbers
msc: [03F30]
defines: 0HNATXFEV5
uses: [0MWKJ44V5A, 0W5DSDFQ4G]
sources:
  - url: https://en.wikipedia.org/wiki/Natural_number
    locator: Order
---

## Definition

Let $a$ and $b$ be [[natural numbers|0MWKJ44V5A]].
The notation $a \le b$ means that there is a natural number $c$ with $a + c = b$, where $+$ is the [[addition|0W5DSDFQ4G]] of $\mathbb{N}$.
The notation $a < b$ means that $a \le b$ and $a \neq b$.

## Remarks

The relation $\le$ is a total order on $\mathbb{N}$: it is reflexive, antisymmetric and transitive, and $a \le b$ or $b \le a$ for all $a$ and $b$.
The order agrees with the operations: if $a \le b$, then $a + c \le b + c$ and $a c \le b c$ for each natural number $c$.
Each nonempty set of natural numbers has a least element for this order. This is the [[well-ordering principle|01BX15HSRR]].

## Explanations

### L1 human

The number $a$ is at most $b$ when you can get from $a$ to $b$ by adding something.
For example, $3 \le 7$ because $3 + 4 = 7$, but $7 \le 3$ is false because no natural number added to $7$ gives $3$.