proof of principle of transfinite induction

To prove the transfinite induction theorem, we note that the class of ordinals is well-ordered by $\in$. So suppose for some $\Phi$, there are ordinals $\alpha$ such that $\Phi(\alpha)$ is not true. Suppose further that $\Phi$ satisfies the hypothesis, i.e. $\forall\alpha(\forall\beta<\alpha(\Phi(\beta))\Rightarrow\Phi(\alpha))$. We will reach a contradiction.

The class $C=\{\alpha:\neg\Phi(\alpha)\}$ is not empty. Note that it may be a proper class, but this is not important. Let $\gamma=\min(C)$ be the $\in$-minimal element of $C$. Then by assumption, for every $\lambda<\gamma$, $\Phi(\lambda)$ is true. Thus, by hypothesis, $\Phi(\gamma)$ is true, contradiction.

Title proof of principle of transfinite induction ProofOfPrincipleOfTransfiniteInduction 2013-03-22 12:29:06 2013-03-22 12:29:06 jihemme (316) jihemme (316) 11 jihemme (316) Proof msc 03B10