Wosets are Isomorphic to Each Other or Initial Segments

Theorem
Let $\struct {S, \preceq_S}$ and $\struct {T, \preceq_T}$ be well-ordered sets.

Then precisely one of the following hold:


 * $\struct {S, \preceq_S}$ is order isomorphic to $\struct {T, \preceq_T}$

or:
 * $\struct {S, \preceq_S}$ is order isomorphic to an initial segment in $\struct {T, \preceq_T}$

or:
 * $\struct {T, \preceq_T}$ is order isomorphic to an initial segment in $\struct {S, \preceq_S}$

Also see

 * Counting Theorem