Axiom:Ordering Axioms/Class Formulation

Definition

Let $V$ be a basic universe.

Let $\RR \subseteq V \times V$ be a relation on $V$.

$\RR$ is an ordering in $V$ if and only if $\RR$ satisfies the axioms:

\((1)\)	$:$		$\RR$ is reflexive	\(\ds \forall a \in \Field \RR:\)	\(\ds a \mathrel \RR a \)
\((2)\)	$:$		$\RR$ is transitive	\(\ds \forall a, b, c \in \Field \RR:\)	\(\ds a \mathrel \RR b \land b \mathrel \RR c \implies a \mathrel \RR c \)
\((3)\)	$:$		$\RR$ is antisymmetric	\(\ds \forall a, b \in \Field \RR:\)	\(\ds a \mathrel \RR b \land b \mathrel \RR a \implies a = b \)

where $\Field \RR$ denotes the field of $\RR$.

These criteria are called the ordering axioms.

Also see

• Axiom:Ordering Axioms/Formulation 1 for an alternative formulation of the ordering axioms on a set.
• Definition:Ordering

• Results about orderings can be found here.