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Transfinite induction has two canonical statements. The "strong" statement, $$(\forall \alpha,\beta)((\beta<\alpha) \wedge P(\beta) \rightarrow P(\alpha))\rightarrow (\forall \alpha)P(\alpha),$$ doesn't split anything into cases. The version used most frequently in proofs says that any property preserved under unions and successors holds for all ordinals. Zero should rarely be a special case.