An odd -- probably basic -- question about model theory:

For $\mathcal{M}$ a structure in a (first-order) signature $\Sigma$, let $\mathcal{M}'$ be the structure in signature $\Sigma\sqcup\lbrace U\rbrace$ -- with $U$ a unary relation -- whose reduct to $\Sigma$ is $\mathcal{M}$, and interprets $U$ as $$ U^{\mathcal{M}'}=\lbrace a: a\text{ is definable in $\mathcal{M}$}.\rbrace$$

Up to the choice of unary relation symbol $U$, this is well defined; moreover, we can iterate this through the ordinals: $$ \mathcal{M}^{(0)}=\mathcal{M}, \quad \mathcal{M}^{(\alpha+1)}=(\mathcal{M}^{(\alpha)})', \quad \mathcal{M}^{(\lambda)}=\bigcup_{\beta<\lambda}\mathcal{M}^{(\beta)} \,\,(\lambda \text{ limit}).$$ (The union notation is technically inappropriate, but its meaning is clear.) Now, for any $\mathcal{M}$, let $$ D(\mathcal{M}, \alpha)=\lbrace a\in\mathcal{M}: a\text{ is definable in $\mathcal{M}^{(\alpha)}$}\rbrace$$ be the set of elements of $\mathcal{M}$ definable after stage $\alpha$. Let $D(\mathcal{M}, \infty)=\bigcup_{\alpha\in ON} D(\mathcal{M}, \alpha)$ be the set of all eventually definable elements, and for $a\in D(\mathcal{M}, \infty)$ let the *age* of $a$, $age_\mathcal{M}(a)$, be the least $\beta$ such that $a\in D(\mathcal{M}, \beta)$. Clearly for each $\mathcal{M}$ there is a least upper bound, $m_\mathcal{M}$, on the ages of elements of $D(\mathcal{M}, \infty)$.

Some very easy observations:

If $\mathcal{M}=(M, <)$ is a well-ordering, then $D(\mathcal{M}, \infty)=M$, since the $\alpha$th element of $\mathcal{M}$ is definable by stage $\alpha$ at the latest.

Even if $\mathcal{M}$ is strongly minimal, $\mathcal{M}'$ need not be: consider $\mathcal{M}=\mathbb{N}+\mathbb{Z}$ as a linear order. Presumably other niceness properties such as stability are also not preserved, but I don't have examples yet.

My question is, what is known about the set $D(\mathcal{M}, \infty)$, the age function $age_\mathcal{M}$, or the invariant $m_\mathcal{M}$? I've been playing around with this idea for a bit, but my model theory is not very strong; I'm sure this has been treated before, but I haven't been able to find a reference.

(In case anyone is interested, I initially thought that there would be connections with notions of rank, as long as $\mathcal{M}$ is sufficiently nice; in fact, I came up with this question after using some dubious analogies to try to explain forking and rank to a friend. As far as I can tell, this initial hope is in fact bogus, but that's where this came from.)

There are two other questions about this that I'm especially interested in. First, what if we augment first-order logic by adding a logical unary relation $D$ whose interpretation is stipulated to always be $D(\mathcal{M}, \infty)$ -- the resulting model theory seems wild (compactness and Lowenheim-Skolem fail extremely badly), but this logic "comes from" first-order logic in a natural way; is there anything nice we can say about it? Second, this time closer to computability theory: what if we replace "definable" with $\Sigma_1$-definable? Does $m_\mathcal{M}$ now have a recursion-theoretic interpretation? I consider these as just curiosities, compared to the main question (which, though more vague, I hope is still appropriate), but if anyone has anything to say on either count I'd be extremely interested.