Well founded relations.

Question

I'm reading a proof in Jech Set theory and I cannot understand a line. Why is it the case that the replacement axiom guarantees the existence of $\theta$ such that $P_\theta = P_{\theta + 1}$? Last line below.

Answer 1

A simple transfinite induction proof shows that $P_\alpha\subseteq P_\beta$ when $\alpha\le\beta$, and it is similarly obvious that $P_\alpha\subseteq P$. If it were true that $P_\alpha\subsetneq P_{\alpha+1}$ for every ordinal $\alpha$, then the function $f:{\sf On}\to {\cal P}(P):\alpha\mapsto P_{\alpha+1}\setminus P_\alpha$ would map each ordinal to one of a collection of disjoint nonempty sets, which are therefore distinct, so this function is one-to-one. Then the inverse function $f^{-1}:\operatorname{ran}f\to{\sf On}$ is a bijection, and $\operatorname{ran}f\subseteq{\cal P}(P)$, so $\operatorname{ran}f$ is a set, and replacement gives that $\sf On$ is a set, a contradiction.