I'm reading Manin et al's "A Course in Mathematical Logic for Mathematicians". There is a definition of a complete theory that I can't square with others I've previouly read. I'll give the relevant definitions below:
Let L1 be the class of first-order languages. Let L be a language in L1. Let $\phi$ be an interpretation of L. Let $T_{\phi}L$ be the set of $\phi$-true formulas [i.e., what I refer to as a theory above], where a $\phi$-true formula is a formula that, given an interpretation, is true for all elements of the universe of discourse that we could assign to the free variables in the formula. The set $T_{\phi}L$ is in fact complete, meaning that for any closed formula P, either P or $\neg$P lies in $T_{\phi}L$.
Now, this sounds like the definition of syntactic completion, except that syntactic completion says that any formula or its negation is provable, and first-order logic is not syntactically complete--if I understand the incompleteness theorem at a layman's level. I haven't found anything to backup the assertion that any formula of a first-order language or its negation is true, nor a definition of this as completeness, so I'm reaching out here.
I am assuming that by
you actually meant
With that said there is nothing wrong with $T_\phi(L)$ being complete.
Indeed the only closed formulas in $T_\phi(L)$ are those which are true in the interpretation $\phi$ and it can be proved by the definition of the semantics for first-order logic languages that every closed formula is either true or false once you fix an interpretation, hence either it will belong to $T_\phi(L)$ or its negation will do.
The fact that $T_\phi(L)$ is complete does not imply in any way that the first-order logic (of language $L$) is complete. In particular $T_\phi(L)$ is not the set of logic formulas which hold in every possible $L$-structure, that is, by the completeness theorem, the set of all formulas provable by first-order logic's inference rules.
So there is no problem with the incompleteness theorems here.