This seems super intuitive, but I can't seem to prove it.
Is the set of equivalence classes of totally ordered sets totally ordered?
More precisely, given two totally ordered sets, $F$ and $G$, does there always exist an order preserving injection from one into the other?
I would think there is some adaptation of Zermelo's theorem that can fix the problem, but again, I can't seem to find it.
Best Answer
The natural numbers, $\omega$ and their inverse order (i.e. the negative integers) are incomparable.
But this is not even a partial order. The rational numbers embed into $[0,1]\cap\Bbb Q$ and vice versa, so it's not even antisymmetric.