Transfinite Recursion
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. Induction by cases Let P(\alpha) be a property defined for all ordinals \alpha. Suppose that whenever P(\beta) is true for all \beta < \alpha, then $P(\backslash alpha)$ is also true. Then transfinite induction tells us that $P$ is true for all ordinals. Usually the proof is broken down into three cases: * Zero case: Prove that $P(0)$ is true. * Successor case: Prove that for any successor ordinal $\backslash alpha+1$, $P(\backslash alpha+1)$ follows from $P(\backslash alpha)$ (and, if necessary, $P(\backslash beta)$ for all ). * Limit case: Prove that for any