Transfinite induction

Representation of the ordinal numbers up to $\omega ^{\omega }$. Each turn of the spiral represents one power of $\omega$. Transfinite induction requires proving a base case (used for 0), a successor case (used for those ordinals which have a predecessor), and a limit case (used for nonzero ordinals which do not have a predecessor).

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.