# transfinite induction

Suppose $\Phi(\alpha)$ is a property defined for every ordinal $\alpha$, the principle of transfinite induction states that in the case where for every $\alpha$, if the fact that $\Phi(\beta)$ is true for every $\beta<\alpha$ implies that $\Phi(\alpha)$ is true, then $\Phi(\alpha)$ is true for every ordinal $\alpha$. Formally :

 $\forall\alpha(\forall\beta(\beta<\alpha\Rightarrow\Phi(\beta))\Rightarrow\Phi(% \alpha))\Rightarrow\forall\alpha(\Phi(\alpha))$

The principle of transfinite induction is very similar to the principle of finite induction, except that it is stated in terms of the whole class of the ordinals.

Title transfinite induction TransfiniteInduction 2013-03-22 12:29:03 2013-03-22 12:29:03 jihemme (316) jihemme (316) 10 jihemme (316) Theorem msc 03B10 principle of transfinite induction PrincipleOfFiniteInduction Induction TransfiniteRecursion