On Fri, Feb 02, 2007 at 03:37:40PM -0500, Conrad Bock wrote:
> Chris,
>
> > Well, yes, of course, models aren't axioms, but facts about the
> > relation between a theory and its intended models can *most
> > definitely* be expressed and proved in a completely formal,
> > axiomatic way. Fact is, though, model theory is just an
> > application of Zermelo-Fraenkel (or whatever) set theory. Just as
> > we can define the numbers to be set theoretic objects, languages
> > can also be represented as set theoretic objects as well (or you
> > can take them to be sui generis and use ZFU, i.e., ZF with
> > urelements). Model structures are themselves set theoretic
> > objects, of course, and all of the usual semantic relations
> > between a language L and an appropriate class of model structures
> > (notably, truth in a structure) defined by an interpretation of L
> > are rigorous mathematical relations. Moreover, it is perfectly
> > possible within this framework to declare a certain class of
> > models of a theory to be intended. For instance, in ZF, we can
> > declare omega-models to be the intended models of PA. This is all
> > *completely* rigorous. For example, we can say with great
> > precision precisely what the structure of all countable, non-omega
> > models of PA look like (N + QZ for logic nerds, where N, Q, and Z
> > are the order-types of the natural numbers, rational numbers, and
> > integers, respectively) -- from which it follows immediately that
> > no such model is intended.
>
> If the models are axiomized under ZF, then the model theory is just a
> mapping from one set of axioms to another, which of course can be
> axiomitized. (01)
Conrad, I don't understand this response. If the model theory of a
language is axiomatized under ZF (I'm not sure it means to axiomatize a
*model*), the model theory is NOT a mapping from one set of axioms to
another. It is a theory, expressed within ZF, of the formal relations
between languages and their interpretations and consequently between
theories and their models. (02)
> This doesn't get you much closer to an assessment of ontological
> correctness, unless you happen to feel comforable with the
> axiomitization of the model. (03)
By "axiomatization of the model" I take you to mean something like
"definition of the class of models" of a given theory. I guess I'm not
sure what the point is here. It is possible of course that one might
define a class of intended models and then, lacking a formal proof of
the fact, doubt whether one's theory picks out exactly that class.
Sure, that might happen. But so what? There are plenty of cases where
we're quite confident of the model theory and hence where various
notions of correctness might have some purchase, e.g., PSL. Indeed, if
you *can't* describe the intended models of a theory mathematically, it
seems to follow that you might not be terribly sure that you know what
your theory is describing in the first place. (This was in fact a real
problem for the untyped lambda calculus until the great Dana Scott came
up with domain theory.) (04)
> If you don't, then there's the "deja vu" effect I was referring to.
> The model theory is just a mapping from one thing you're not sure of
> to another. (05)
I didn't get the deju vu metaphor, but if you've *got* a model theory
for a language in which you've characterized a class of intended models,
you *have* to have described those models in a way you are sure of. Of
course, as noted, you might doubt whether the class you've picked out is
the class you really *want* (e.g., lacking a proof, you might not be
sure whether your axioms characterize exactly that class), but all that
seems to mean is that you have some work to do. (06)
-chris (07)
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