I would like to know if the set of natural numbers is the only one where the well-ordering principle holds for the usual order relation ?
I have troubles understanding this
Thanks for your answers!
2026-04-05 19:38:16.1775417896
Is the well-ordering principles only holding for the natural numbers?
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If you only want to use the usual order relation, the sets $$\{-k,-k+1,-k+2,...,0,1,2,3...\}$$ where $k\in\mathbb{N}$ also satisfy the well-ordering principle. It is order-isomorphic to $\mathbb{N}$ so it's not really a great example.
If you're willing to use other order relations, then any set can be well-ordered according to the axiom of choice. They can be made explicit on same cases (for instance in $\mathbb{Z}$) but usually not.