Alternative Definitions of Complete Theories 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. 
 A: I am assuming that by 

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.

you actually meant 

where a $\phi$-true formula is a formula that, given the interpretation, is true for all elements of the universe of discourse that we could assign to the free variables in the formula.

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.
