# Contents

## Definition

In a strictly ordered ring $R$, the natural numbers are a subset of $R$, because every strictly ordered ring has characteristic zero. A number or element $a \in R$ is an infinite number or an infinite element if for all natural numbers $n \in \mathbb{N}$, $n \lt a$.

Examples of rings with infinite numbers include the hyperreal numbers and the surreal numbers.

A strictly ordered ring $R$ which satisfies the archimedean property has no infinite numbers.