[FOM] Rigorous Foundations of Model Theory?

[Rex Butler]

As an enthusiast of Foundations of Mathematics, I have yet to see a
clear cut declaration of what exactly the rigorous foundations of
Model Theory are.  For example, if I was to codify model theory in a
computer verification system, how might I start?  My confusion comes
from deflections of the issue I have seen in the literature such as
the following:  "Let A be a set.  R is an n-ary relation over A (n >=
1) if R subset A^n; that is, for all a_1,...,a_n it is in some way
determined whether the statement that R(a_1,...,a_n) is true or
false." --- this example being from Basic Model Theory by Kees Doets,
though I am sure this is not the only case of vague language.

Surely 'it is in some way determined' is an exceptionally nebulous
statement for an exacting subject as FOM, and I'm guessing this issue
is related to the notion of 'sacred' vs 'profane' versions of
foundational studies.

My guess as to one approach is as follows: after the 'foundational
aspects' such as completeness, etc... which set the enterprise on a
sure footing, we treat Model Theory like any other subject, codify
first order logic within ZFC (say) and treat models just like vector
spaces, as a set theoretic construction alongside this 'internal'
first order logic.  Though having ZFC as a meta theory is much too
strong for some, I would assume, which accounts for the lack of
commitment in model theory texts.

Am I heading in the right direction?

Thanks,

Rex Butler
MS Mathematics
University of Utah