Well-order - Equivalent Formulations

Equivalent Formulations

If a set is totally ordered, then the following are equivalent to each other:

  1. The set is well-ordered. That is, every nonempty subset has a least element.
  2. Transfinite induction works for the entire ordered set.
  3. Every strictly decreasing sequence of elements of the set must terminate after only finitely many steps (assuming the axiom of dependent choice).
  4. Every subordering is isomorphic to an initial segment.

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