Definition of truth in first-order logic Let $L$ be a first order language.
Let $P$ be a predicate symbol from $L$, and $c$ a constant. Given an interpretation $I$ of $L$, a definition states

The formula  $P(c)$ is true in $I$ iff $c\in I(P)$.

My question then is, what does it mean to say that ' "$c\in I(P)$" is true'? This is not a formula in $L$, it could be regarded a formula in an augmented language $L'\supset L$, but then I could ask the same question about $L', I'$.
I hope my question makes sense.
 A: The statement $c\in I(P)$ is in the metatheory, so it is assumed you already know what it means.  That is, assuming you have some background set theory in which you can talk about statements like $c\in I(P)$ and determine whether they are true, you are defining a new concept called "truth" about formulas and structures for your first-order language.  You are not trying to define mathematical "truth" in any absolute sense, just in this very particular context of first-order structures.
If you want to say what it means for $c\in I(P)$ to be true, that's more a matter of philosophy than mathematics.  Typically we treat the "truth" of things like this as an undefined notion: we just have formal rules for how we can make logical deductions among statements like this (if these statements are true, then we can conclude that those other statements are true).  But we don't normally give any actual definition of "truth" for statements in the metatheory.  If you're a Platonist, you might imagine that there actually is a real set-theoretic universe that exists in some sense, and then "truth" is an empirical assertion about this universe (it's saying that $c$ really is an element of the set $I(P)$).  But this is philosophy, not mathematics.
A: Generally in logic when you talk about a true statement is because you have a particular model in mind that is your relative truth.
When axiomatizing the truth, you aim to make the most precise definition you can make, in the sense that no other model but your choice of true model satisfies your axioms. Gödel's results show you that this task cannot always be accomplished.
For example, it is widely understood in arithmetic than when you talk about true statements you are actually referring to statements true in the standard model, which only contains the standard numbers.
In your example, I take that you are defining a particular model to be your truth via a syntactical correspondence.
