[FOM] Rigorous Foundations of Model Theory?

[Monroe Eskew]

If you have clear definitions for the basic notions of model theory,
that should suffice for a rigorous foundation.  The notions can be
defined within ZF(C) set theory, but other set theories can probably
make sense out of model theory too.

A standard treatment starts with a language L as a collection of
constant symbols, relation and function symbols of various finite
arity.  One also has fixed symbols for conjunction, negation,
quantifier, parentheses, variables.  One defines recursively what
counts as a term and a formula.  An L-model M is an ordered pair
(A,f), where A is the universe of the model, and f is a function
assigning (a) to each constant symbol, an element of A, (b) to each
n-ary relation symbol, a subset of A^n, (c) to each n-ary function
symbol, a function from A^n to A.  One then defines by induction on
complexity of formulas what it is for a model M to satisfy a formula.
To answer your question, if M assigns to the n-ary relation symbol R a
subset R' of A^n, then M satisfies R(a_1,...,a_n) iff (a_1,...,a_n) is
a member of R'.

In some set theories like Morse-Kelley, you can have proper-class
models.  There might even be a reasonable way to interpret model
theory within axiomatic category theory, for all I know.

On Mon, Oct 5, 2009 at 6:01 PM, Rex Butler <rexbutler at gmail.com> wrote:
>
> 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
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