Re: question about categoricity

Torkel Franzen wrote:
> "Per Freem" <perfreem@xxxxxxxxx> writes:
>
> > i see, yes i meant to make that assumption about the constants--i
> > apologize for not making it clearer. the reason i am trying to find
> > types omitted by one model of a complete theory and realized by another
> > to determine nonisomorphism is because the 'model existence proofs'
> > unfortunately do not make sense to me.
>
> I don't see just what your problem is. Are you trying to prove a
> general statement? In the case of the two structures given by Chris
> Menzel, you can prove directly that they are not isomorphic by just
> applying the definition of isomorphism. As for types, one model
> realizes the (non-principal) type {not x = d0, not x = d1,...} and
> the other does not.

ok this is exactly what i was looking for, i had trouble making the
leap from the structures given by Chris to an actual type. but now that
you've given the type, i'm assuming d0, ..., dn are constants. but
since Chris's example involves expanding the language of the first
structure (by adding another constant to it), for the proof to work you
must eventually take the reduct of that structure to your original
language, which does not have the problematic constant k which has this
funny property. so why would this type be realized? it seems
paradoxical to me. if we posit this new constant k that has a property
like being larger than all the other constants, and come up with an
appropriate type to pick it, then if the type is realized, then there
is something in the domain satisfying the condition (something larger
than all the rest of the things denoted by constants--since we assumed
we have constants to name everything in the domain) -- and i understand
that in /that/ case, the model you have with the extra constant will
not isomorphic to the model you started with. but since this constant
has to denote something larger than the rest, it means your domain is
uncountable. but then lowenheim-skolem pushes you back down to say that
there's a countable structure--but how do you know the non-isomorphism
is not lost then? after all you have to drop down to a language
/without/ that special additional constant. this is what is frustrating
me about this proof--do you know what i'm missing?

appreciate your patience and help, looking forward to hearing from you.

p.s. i of course don't mean to suggest that i'm onto some paradox; it
seems paradoxical to me because i am clearly missing something, i don't
actually doubt that there /is/ an elementarily equivalent countable
model of the theory, say, T = Th((Nat, <)) that is not isomorphic to
it.

.

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