Ordered field

An ordered field is a field $F$ (Field) together with a subset $P\subseteq F$, the set of positive elements (the positive cone), satisfying:

• (O1) Trichotomy. For each $x\in F$, exactly one of the following holds: $x\in P$, or $x=0$, or $-x\in P$.
• (O2) Closure. If $x,y\in P$ then $x+y\in P$ and $x\cdot y\in P$.

We write $x>0$ for $x\in P$ (read "$x$ is positive"), and define the order by

$a<b:⟺b-a\in P,a\le b:⟺a<b\text{ or }a=b\mathrm{.}$

Thus $a>b$ means $b<a$, and $a\ge b$ means $b\le a$. An element $x$ with $-x\in P$ (equivalently $x<0$) is called negative.

Field

• By trichotomy applied to $b-a$, for any $a,b$ exactly one of $a<b$, $a=b$, $a>b$ holds; this makes $\le$ a total order.
• (O2) says the positives are closed under addition and multiplication: sums and products of positives are positive.
• The rationals (The rationals form a totally ordered field) and both constructions of the reals (The reals form a totally ordered field, The Dedekind reals form a totally ordered field) are ordered fields, so every fact proved here from (O1)-(O2) holds in each of them.