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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 ordinals which don't have a predecessor).

Transfinite induction is an extension of mathematical induction to well-ordered sets, for example to sets of ordinal numbers or cardinal numbers. Its correctness is a theorem of ZFC.^[1]

^ J. Schlöder, Ordinal Arithmetic. Accessed 2022-03-24.