= Solution
For a <set> $M$, let $\operatorname{Def}(M)$ consist of the <subsets> of $M$ definable over the structure $\langle M,\in\rangle$ 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 $M\subseteq\operatorname{Def}(M)$.
Define the <constructible hierarchy> by <transfinite recursion>:
$$
L_0=\varnothing,\qquad L_{\alpha+1}=\operatorname{Def}(L_\alpha),\qquad
L_\lambda=\bigcup_{\alpha<\lambda}L_\alpha\quad(\lambda\text{ a nonzero limit ordinal}).
$$
Gödel's <constructible universe> is the definable <class in set theory>
$$
\boxed{L=\{x:\exists\alpha\text{ ordinal }(x\in L_\alpha)\}.}
$$
At 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 $\operatorname{Def}$.
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