Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 128 / 1 / b / Solution

Work in the ambient universe and let $a,u\in L$. SupposeFor each $x\in a$, let $y_x$ be this unique witness. The relativization ${\phi }^L$ is a first-order formula, so the ambient Axiom schema of replacement collects the witnesses $y_x$ into a set. Every witness lies in the constructible hierarchy, hence there is an ordinal $\alpha$ such thatFor example, take the supremum of one constructible rank for each witness and then increase it by one.

The set $L_{\alpha }$ itself belongs to $L_{\alpha +1}\subseteq L$. Taking $b=L_{\alpha }$, every $x\in a$ has a witness $y\in b$ satisfying ${\phi }^L(x,y,u)$. Thereforewhich is the stated instance of Replacement.