# Isomorphism Class of Total Orderings

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## 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.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{ProofWanted}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

## Sources

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