# Definition:Extended Natural Numbers

## Definition

Let $\N$ be the set of natural numbers.

### Definition 1

The extended natural numbers is the ordered set $(\N \cup \{\infty\}, \leq)$, where

$\infty$ is a new element, called infinity
$\leq$ is the ordering on the extended natural numbers

### Definition 2

The extended natural numbers are the order completion of the ordered set of natural numbers.