Definition:Well-Ordering
Definition
Let $\struct {S, \preceq}$ be an ordered set.
Definition 1
The ordering $\preceq$ is a well-ordering on $S$ if and only if every non-empty subset of $S$ has a smallest element under $\preceq$:
- $\forall T \subseteq S, T \ne \O: \exists a \in T: \forall x \in T: a \preceq x$
Definition 2
The ordering $\preceq$ is a well-ordering on $S$ if and only if $\preceq$ is a well-founded total ordering.
Class Theory
In the context of class theory, the definition follows the same lines:
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 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.