 Solution

For a set $M$, let $\mathrm{Def}(M)$ consist of the subsets of $M$ definable over the structure $⟨M,\in ⟩$ by a first-order formula with finitely many parameters from $M$. It is a set contained in $\mathcal{P}(M)$: satisfaction for a set structure and the set of logical formula codes give a uniform set construction of these definable subsets. If $M$ is transitive, every $a\in M$ is one such subset, defined over $M$ by $z\in a$, so $M\subseteq \mathrm{Def}(M)$.

Define the constructible hierarchy by transfinite recursion:Gödel's constructible universe is the definable class in set theoryAt successor stages only subsets definable over the previous stage are admitted, in contrast to the full power-set construction of the cumulative hierarchy. The stages are transitive and increasing, as permitted in the question, and their union contains every ordinal. Finite parameters are essential in the definition of $\mathrm{Def}$.