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Well-order
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== Equivalent formulations == If a set is [[Total order|totally ordered]], then the following are equivalent to each other: # The set is well ordered. That is, every nonempty subset has a least element. # [[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.
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