On Thu, Mar 09, 2006 at 01:55:58PM -0500, Michael Gruninger wrote:
> ...the methodology is:
>
> 1. Specify a class of structures
> 2. Existence Theorem : show that this class is nonempty
> 3. Classification Theorem: classify the structures in this set up to
> isomorphism
> 4. Show that each structure in the class is a model of the theory
> 5. Show that each model of the theory is elementarily equivalent
> to a structure in the class. (01)
That's nice and clear. (02)
> Pedantically speaking, these are relative consistency proofs, since
> the structures are typically defined with respect to classes of
> graphs, geometries, ordered sets, groups, semigroups etc. which we
> assume exist and are nonempty. (03)
Well, yeah, when we're talking about interesting theories, "consistent",
of course, typically means "can be proved in ZFC to have a model". :-) (04)
> The classification theorem is not always necessary, but it does give
> the best "picture" of the possible models of the theory, alerting
> people to models that are potentially unintended. It also gives more
> insight into the theory. For example, you could say that a group is
> any structure that satisfies the group theory axioms, but this doesn't
> tell us much; (05)
Indeed, doesn't even tell you there *are* any. (06)
> you get much more sense of the models from the classification theorem
> and structure theorems that tell you how to decompose groups into
> subgroups. (07)
Agreed, of course! (08)
On a purely point of technical fun, I'm not sure why you call step 5
"Axiomatizability". Let me follow your methodology. My theory is True
Arithmetic (i.e., the set of sentences of number theory true in the
standard model). (09)
Step 1: My class is the class N* of structures isomorphic to the
standard model (or, more concretely, to omega with the usual definitions
of ordinal addition and multiplication). (010)
Step 2: ZF proves this class is nonempty (by the axioms of infinity and
separation). (011)
Step 3: Done, by definition of N*. (012)
Step 4: Done, by the definitions of True Arithmetic and N*. (013)
Step 5: Done, by the completeness of True Arithmetic and the definition
of elementary equivalence. (014)
BUT: True Arithmetic is not (recursively) axiomatizable, by Gödel. (015)
-chris (016)
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