 Relative constructible universe

For a transitive set $X$, the relative constructible universe $L(X)$ is the union of the relative constructible hierarchy, starting at $L_0(X)=X$, applying the definable power set at successors and taking unions at limit ordinals. It is the smallest inner model of ZF containing $X$ and all its members. For an arbitrary set $A$, take the base $X$ to be the transitive closure of $\{A\}$, so that $A\in X$. Axiom of choice need not hold for an arbitrary base; the construction uses initial parameters rather than a predicate relativization.