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.

$\blacksquare$