# Definition:Extended Natural Numbers

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## Definition

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

### Definition 1

The **extended natural numbers** are the elements of the ordered set $\struct {\N \cup \set \infty, \le}$, where

- $\infty$ is a new element, called
**infinity** - $\le$ 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.