Definable closure in class-sized expansions of o-minimal groups I am working in NBG set theory with limitation of size (i.e. the class of all sets is in bijection with the class of ordinals).
Let $\mathbf{G}$ be a class-sized o-minimal expansion of an ordered group. By o-minimal, I mean that every definable subclass of $\mathbf{G}$ is a finite union of intervals. Let $X \subset \mathbf{G}$ be a subset. It can be shown that the definable closure $\operatorname{dcl}(X)$ of $X$ is an elementary substructure of $\mathbf{G}$.
Well, this is not really true, and this is the gist of my question. The problem is that it is not clear to me that the definable closure exists as per NBG set theory. If $\mathbf{G}$ is a real-closed field, with no additional structure, then the definable closure (i.e. relative real closure) can be described, and its existence can be shown using the class comprehension scheme theorem of NBG set theory. In certain cases (e.g. elementary extensions of the real exponential field with restricted analytic functions), $\emptyset$-definable maps can be described, and the definable closure exists for similar reasons.
My question is: does $\operatorname{dcl}(X)$ exist in the general case of o-minimal expansions of ordered groups?
 A: I am not sure exactly how you intend your question, but I am interpreting on very general grounds, concerning the circumstances in which NGB is able to provide the notion of definability and truth for class-sized structures.
The main observation is that one cannot generally refer to truth in class-sized structures in NGB, because one cannot prove that there is a Tarskian truth predicate. For example, Tarski's theorem on the nondefinability of truth shows that there can be no definable satisfaction class or truth predicate for the whole structure $\langle V,\in\rangle$, and so in NGB, there may be no such satisfaction class at all. And in general, other structures such as groups and graphs can be interpretatively just as strong, which would prevent them also from having a truth predicate.
The main obstacle in play is that the recursive definition of the satisfaction relation is not a set recursion, but a class recursion — one defines the class-sized truth predicate for a formula by reference to the truth predicates for  its immediate subformulas. Such a recursion cannot in general be undertaken in NGB.
Meanwhile, in the slightly stronger Kelley-Morse set theory, however, every class structure does have a truth predicate. And in fact, one needs not full KM but only the principle ETR, which allows for class recursions, and indeed only $\text{ETR}_\omega$, since the definition of satisfaction is a recursion of length $\omega$, recursion on formulas.
Without the truth predicate, one cannot refer to definability in general or speak of the definable closure.
In special cases where the structures are particularly simple, for example, if they omit quantifiers and what not, then you will get a definable satisfaction class in NGB. Your o-minimality requirement essentially provides this. In the case of the real-closed fields, you can observe the QE result that shows that with Q-free satisfaction (which requires no recursion) you can define a satisfaction class and thereby get the notion of definable closure.
I am a little less sure with o-minimality, since one needs to know about the higher-dimensional analogues. I am weaker on this model-theoretic point than I would like. If you have uniformly definable truth for those higher-dimensional subsets, however, then this should be enough for you to be able to define a satisfaction class and get the notion of definable closure.
