This is my proof of Transfinite Induction by filling the gaps in my textbook. It would be great if someone help me verify if i correctly understand what is meant by the authors!
Let $P(\alpha)$ is a property defined for all ordinals $\alpha$. Suppose that $P(\alpha)$ is true for all $\alpha<\beta$ implies $P(\beta)$ is also true. Then $P(\alpha)$ is true for all ordinals $\alpha$.
My attempt:
Assume the contrary that $P(\gamma)$ is not true for some ordinal $\gamma$.
Let $F:=\{\alpha \in\gamma\cup\{\gamma\}\mid P(\alpha) \text{ is not true}\}$.
It follows that $\gamma\in F$ and thus $F\neq\emptyset$, and that $F$ is a set of ordinals and thus is well-ordered.
Let $\beta=\min F$. Then $\alpha\notin F$ for all $\alpha<\beta$. This implies that $P(\alpha)$ is true for all $\alpha<\beta$. By inductive hypothesis, $P(\beta)$ is also true and thus $\beta\notin F$. This is a contradiction.
Hence $P(\alpha)$ is true for all ordinals $\alpha$.