transfinite induction

Suppose $\mathrm{\Phi }(\alpha )$ 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 $\mathrm{\Phi }(\beta )$ is true for every implies that $\mathrm{\Phi }(\alpha )$ is true, then $\mathrm{\Phi }(\alpha )$ is true for every ordinal $\alpha$. Formally :

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.