Axiom:Ordering Axioms/Class Formulation
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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.