TITLE: What is gained by internalizing LST (the language of set theory)?
QUESTION [6 upvotes]: I'm reading up on Gödels constructible universe L in the book "Constructibility" by Devlin, and by comparing his text with texts like Kunen and Jech, there is one thing in particular that he's doing differently: internalizing LST (that is, the language $\{\in\}$) inside set theory itself.
To give you a brief picture of what he does, his construction is roughly to interpret every logical symbol (connectives, variables, parentheses) as natural numbers inside set theory itself, and formulas as sequences of natural numbers, again inside set theory. He proceeds to prove that on transitive sets, this interpretation coincides with the meta-mathematical one via LST (for $\Delta_0$ formulas; that is, formulas where every quantifier is bounded). He spends an awful lot of time on this, and I cannot seem to see what he gains from such a construction.
My question is then what such an internalized construction achieves? Is it so that theorems about formulas of ZFC can "make more sense" - that no need of meta-mathematical theorems is needed? Mere convenience? Philosophical issues? It doesn't need to specifically be about Devlin's construction, but more about an arbitrary internalization of set theory.
I apologize in advance if this question is too vague, but I tried to make it as specific as possible.

REPLY [2 votes]: The main point is the need to quantify over formulae.
I.e., we need to say "there exists a formula such that ...".
Or "For every formula it holds that ..."
The problem is that in set theory we can quantify only over SETS.
So, we cheat, so to speak.
We find a collection of sets which behave like the formula.
Now we can quantify over these sets.
In order for this to be useful there should be a connection between the real formula and the sets coding this formula.
Theorems like I.9.11  and I.9.15 in Devling's book are examples of such connection.