It's quite easy to show that a finite set is well ordered iff it is totally ordered. Is the converse also true? That is: is it true that a set is infinite iff it admits a total order which is not a well order? (For the sake of brevity, I shall write t.o. and w.o. for total order and well order, respectively)
The if part follows from the previous observation (t.o.+finiteness implies w.o.), so the question becomes
Is it true that a set is infinite only if it admits a total order which is not a well order?
I know that every set admits a w.o. (and thus a t.o.); still, answering the question requires to prove (or to disprove) that t.o.≠w.o., and this is out of my reach.
The question supposes ZFC but every other set theory is accepted.