Order axioms (for R as an ordered field)

The order axioms for $\mathbb{R}$ assert that there is a total order $\le$ such that:

• (Trichotomy) for all $a,b\in\mathbb{R}$, exactly one of $a<b$, $a=b$, $a>b$ holds,
• (Transitivity) $a\le b$ and $b\le c$ imply $a\le c$,
• (Compatibility with addition) if $a\le b$ then $a+c\le b+c$ for all $c$,
• (Compatibility with multiplication) if $0\le a$ and $0\le b$ then $0\le ab$.

Together with the field axioms , these make $\mathbb{R}$ an ordered field, enabling inequalities, monotonicity arguments, and notions like boundedness and supremum .