Equivalence of Definitions of Strictly Well-Ordered Set
Theorem
The following definitions of the concept of Strictly Well-Ordered Set are equivalent:
Definition 1
Let $\struct {S, \RR}$ be a strictly totally ordered set.
Then $\struct {S, \RR}$ is a strictly well-ordered set if and only if $\RR$ is a strictly well-founded relation.
Definition 2
Let $\struct {S, \RR}$ be a relational structure.
Let $\RR$ be a strict well-ordering on $S$.
Then $\struct {S, \RR}$ is a strictly well-ordered set.
Proof
By definition, a strictly totally ordered set is a relational structure such that $\RR$ is a strict total ordering.
By definition, a strict well-ordering $\RR$ is a strict total ordering such that $\RR$ is a strictly well-founded relation.
So $\struct {S, \RR}$ is a relational structure such that $\RR$ is a strict well-ordering on $S$.
Hence both definitions specify:
- a relational structure $\struct {S, \RR}$
such that:
- $\RR$ is a strict total ordering
- $\RR$ is a strictly well-founded relation.
$\blacksquare$