Isomorphism Class of Total Orderings

Theorem

Let $S$ be a finite set with $n$ elements.

There is exactly one isomorphism class containing the total orderings on $S$.

That is, every total ordering on $S$ is (order) isomorphic to every other total ordering.


Proof


This theorem requires a proof.
In particular: Intuitively obvious, probably needs to be hammered out in a proof by induction. I want to move on to other stuff today.
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Sources

  • 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 14$: Orderings: Exercise $14.5$
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