Transfinite induction

Transfinite induction is an extension of that applies over transfinite ordinals. It can be used in proving the statements about comparisons of fast-growing functions.

Definition
Let \(P(\alpha)\) be a property defined on all ordinals \(\alpha \in \text{On}\). Suppose that if \(P(\beta)\) is true for all \(\beta < \alpha\), then \(P(\alpha)\) is also true. Then transfinite induction tells us that \(P\) is true for all ordinals.

There are three cases to prove:


 * Zero case: prove that \(P(0)\) is true.
 * Successor case: prove that \(\forall \alpha = \beta+1 : P(\alpha) \rightarrow P(\alpha+1)\) (for any successor \(\alpha\), P(\alpha) implies P(\alpha+1).)
 * Limit case: prove that \(\forall \lambda \neq \beta+1 \land \beta < \lambda : P(\beta) \rightarrow P(\lambda)\) (for any limit \(\lambda\), P(\beta) implies P(\lambda).)