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

$a < b \;:\Longleftrightarrow\; b - a \in P, \qquad a \le b \;:\Longleftrightarrow\; a < b \text{ or } a = b.$

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.

• 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.