The ordinals satisfy the principle of transfinite induction: suppose that \(C\) is a class of ordinals such that whenever \(C\) contains all ordinals \(\beta\) smaller than some ordinal \(\alpha\), then \(\alpha\) is also in \(C\). Then the class \(C\) contains all ordinals. Using transfinite induction one can prove in ZFC (and one needs the axiom of Replacement) the important principle of transfinite recursion, which says that, given any definable class-function \(G:V\to V\), one can define a class-function \(F:ON\to V\) such that \(F(\alpha)\) is the value of the function \(G\) applied to the function \(F\) restricted to \(\alpha\). One uses transfinite recursion, for example, in order to define properly the arithmetical operations of addition, product, and exponentiation on the ordinals.