Transfinite induction

Transfinite induction is an extension of mathematical induction to ordinal numbers. Its correctness is a theorem of ZF, and relies on the fact that the ordinal numbers are well-ordered, and thus a statement that is not universally true for all ordinals must have a minimal counterexample. In fact, this principle is also true for arbitrary well-ordered sets, but since any well-ordered set can be indexed by ordinals in an order-preserving way, it suffices to establish the principle for ordinals.

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