Definition:Well-Ordering/Class Theory
Definition
Let $V$ be a basic universe.
Let $\RR \subseteq V \times V$ be a total ordering.
Then $\RR$ is a well-ordering if and only if:
- every non-empty subclass of $\Field \RR$ has a smallest element under $\RR$
where $\Field \RR$ denotes the field of $\RR$.
Also known as
There is a school of thought that suggests that this may be referred to as a strong well-ordering, keeping the terminology different from that of a well-ordering.
The difference is that for an ordering to be a well-ordering, every (non-empty) subset must have a smallest element, whereas for an ordering to be a strong well-ordering, not only every subset but in fact every subclass must have a smallest element.
Also defined as
1955: John L. Kelley: General Topology uses the term well-ordering to mean what $\mathsf{Pr} \infty \mathsf{fWiki}$ calls a strict strong well-ordering.
1980: Kenneth Kunen: Set Theory: An Introduction to Independence Proofs uses the term well-ordering to mean what $\mathsf{Pr} \infty \mathsf{fWiki}$ calls a strict well-ordering.
Also see
- Results about well-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