## Definition ## ### With strict order ### An __ordered field__ is a [[Heyting field]] $(A, +, -, 0, \cdot, 1, #)$ with * a [[strict order]] $\lt$ * a term $s:0 \lt 1$ * two families of dependent terms $$a:A, b:A \vdash \alpha(a, b): (0 \lt a) \times (0 \lt b) \to (0 \lt a + b)$$ $$a:A, b:A \vdash \mu(a, b): (0 \lt a) \times (0 \lt b) \to (0 \lt a \cdot b)$$ ### With positivity ### An ordered field is a [[commutative ring]] with a [[predicate]] $\mathrm{isPositive}$ such that * zero is not positive: $$\mathrm{isPositive}(0) \to \emptyset$$ * one is positive: $$\mathrm{isPositive}(1)$$ * for every term $a:A$, if $a$ is not positive and $-a$ is not positive, then $a = 0$ $$\prod_{a:A} ((\mathrm{isPositive}(a) \to \emptyset) \times (\mathrm{isPositive}(-a) \to \emptyset)) \to (a = 0)$$ * for every term $a:A$, if $a$ is positive, then $-a$ is not positive. $$\prod_{a:A} \prod_{b:A} \mathrm{isPositive}(a) \to (\mathrm{isPositive}(-a) \to \emptyset)$$ * for every term $a:A$, $b:A$, if $a$ is positive, then either $b$ is positive or $a - b$ is positive. $$\prod_{a:A} \prod_{b:A} \mathrm{isPositive}(a) \to \left[\mathrm{isPositive}(b) + \mathrm{isPositive}(a - b)\right]$$ * for every term $a:A$, $b:A$, if $a$ is positive and $b$ is positive, then $a + b$ is positive $$\prod_{a:A} \prod_{b:A} \mathrm{isPositive}(a) \times \mathrm{isPositive}(b) \to \mathrm{isPositive}(a + b)$$ * for every term $a:A$, $b:A$, if $a$ is positive and $b$ is positive, then $a \cdot b$ is positive $$\prod_{a:A} \prod_{b:A} \mathrm{isPositive}(a) \times \mathrm{isPositive}(b) \to \mathrm{isPositive}(a \cdot b)$$ * for every term $a:A$, if $a$ is positive, then there exists a $b$ such that $a \cdot b = 1$ and $b \cdot a = 1$ $$\prod_{a:A} \mathrm{isPositive}(a) \to \left[\sum_{b:A} (a \cdot b = 1) \times (b \cdot a = 1)\right]$$ ## Examples ## * The [[rational numbers]] are a ordered field. ## See also ## * [[ordered integral domain]] * [[Heyting field]] * [[Archimedean ordered field]]