Definition:Total Ordering/Class Theory
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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.
- Results about total orderings can be found here.
- 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