Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 121 / 2 / ii / Solution

Fix the base convention $L_0(X)=X$. For a transitive set $X$, the relative constructible hierarchy isHere the definable power set isThe satisfaction for a set structure in this definition is a definable set-theoretic relation, obtained by finite syntax coding and recursion on a first-order formula. The transfinite recursion therefore defines a class in ZF.

Each level is a transitive set. For transitive $A$, every $a\in A$ is a subset of $A$ definable using the parameter $a$, so $A\subseteq \mathrm{Def}(A)$; also $A\in \mathrm{Def}(A)$ using the always-true first-order formula. Thus the levels increase and $X\in L_1(X)$. Starting instead with $X\cup \{X\}$ is another common indexing convention, but it is not the convention used here. The relative universe need not satisfy axiom of choice: arbitrary $X$ need not have an internally available well-order.