If two sets are well ordered, then one of them is isomorphic to the subset of another.
Is the same true for totally (or linearly) ordered sets? I think it is probably not true, but I struggle to find counterexamples of it.
So, my goal is: to find two totally ordered sets such that neither of them is isomorphic to the subset of another.
Or can we find more than two ($\geq 3$) of such sets? i.e. finding three totally ordered sets, such that none of them is isomorphic to the subsets of any one of the other two.
My thought is pretty restrictive at the moment because the only totally ordered set I can think of now is subsets of $\mathbb R$, which doesn't help solve the problem.