The axiom of choice tells us every set can be well-ordered and an easy exercise is that every partial-order can be extended to a linear-order and a linear-order on a set is a well-order iff any of the following (equivalent) conditions hold:
- Transfinite induction works for the entire ordered set.
- Every strictly decreasing sequence of elements of the set must terminate after only finitely many steps (assuming the axiom of dependent choice).
- Every subordering is isomorphic to an initial segment.
But when (and how) can a linear order be extended to a well-order, I'm guessing that one of the conditions must be a bottom element, any other conditions?
Thanks in advance, any feedback is appreciated.