Chris, (01)
> 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
> ZermeloFraenkel (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 omegamodels 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, nonomega models of PA look like (N + QZ
> for logic nerds, where N, Q, and Z are the ordertypes of the
> natural numbers, rational numbers, and integers, respectively) 
> from which it follows immediately that no such model is intended. (02)
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. 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. 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. (03)
Conrad (04)
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