L(R) in the Solovay collapse extension

Let $R={\mathbb{R}}^{V[G]}$. Define the relativized hierarchy

$L_0(R)=\mathrm{tc}(\{R\}),L_{\alpha +1}(R)=\mathrm{Def}(L_{\alpha }(R),\in ,R\cap L_{\alpha }(R)),$

with unions at limits, and put $L(R)={\bigcup }_{\alpha \in \mathrm{O}\mathrm{r}\mathrm{d}}{L}_{\alpha }(R)$. Equivalently it is the least transitive inner model containing every ordinal and every ambient real. It therefore has exactly the ordinals and reals of $V[G]$.

The hierarchy is definable from the class $R$, and each real parameter belongs to $S$. Induction on $\alpha$ shows that every hierarchy element and every member of its transitive closure is definable from finitely many ordinals and members of $S$. Hence $L(R)\subseteq HOD(S)=M$. The reverse inclusion and the equalities with $HOD(R)$ are not asserted.