Definition:Total Ordering/Class Theory

Definition

Let $V$ be a basic universe.

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

Let $\RR$ be such that:

$(1): \quad \RR$ is an ordering on $\Field \RR$

$(2): \quad \forall x, y \in \Field \RR: x \mathop \RR y \lor y \mathop \RR x$ (that is, $x$ and $y$ are comparable)

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

Then $\RR$ is a total ordering.

Also known as

Some sources refer to a total ordering as a linear ordering, or a simple ordering.

If it is necessary to emphasise that a total ordering $\preceq$ is not strict, then the term weak total ordering may be used.

Also see

• Definition:Totally Ordered Class

• Results about total orderings can be found here.

Sources

• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $4$: Superinduction, Well Ordering and Choice: Part $\text I$ -- Superinduction and Well Ordering: $\S 1$ Introduction to well ordering