A method of proof generalizing mathematical induction, i.e., induction on the natural numbers, to the case of all transfinite ordinals. Mathematical induction may be characterized by the scheme that if the following two clauses can be demonstrated (where ‘ordinal number’ is interpreted as ‘natural number’):

1. 1 $0$ has property $P$.

2. 2 If an ordinal number $\alpha$ has property $P$, then its successor $\alpha +1$ has property $P$

then one can infer that all natural numbers have the property $P$. Transfinite induction adds the additional clause to cover limit ordinals:

1. 3 If $\beta$ is a limit ordinal and all ordinal numbers $\alpha <\beta$ have property $P$, then $\beta$ has property $P$.

A proof by transfinite induction will infer from proofs of Clauses 1–3 that all ordinal numbers have property $P$.