Re: question about categoricity
- From: "Per Freem" <perfreem@xxxxxxxxx>
- Date: 25 Nov 2005 19:42:03 -0800
george wrote:
> You can add INFINITELY many constants and still remain the size of Nat.
> There are uncountably many different ways of making Nat ORDINALLY
> bigger,
> while adding no more than "Nat-many" new things to it.
....
> Chris's answer didn't even address your original question , which was,
> "what stops us from having an isomorphism between them?"
> The short answer is that, obviously, SINCE THEY ARE THE SAME
> CARDINALITY,
> NOTHING can stop you from having an isomorphism between them, if all
> you are
> thinking about is the individuals in the domain. The DEFINITION of
> what it MEANS
> for the domains to be "the same cardinality" is that there DOES exist
> an isomorphism
> between them. For the models to be non-isomorphic, there has to be
> some "other"
> thing (besides the domains) about them that is structured in two
> different ways under
> the two different models.
>
> The decision to invoke "=" here was, as ususal, a mistake.
> You don't need to use = to force the existence of infinitely many
> different things.
....
> This is just illegitimately conceived to begin with.
> There Is No Such Thing As THE theory that claims
> that R's extension is infinite. There are A GREAT MANY DIFFERENT
> ways of dssiging theories claiming that. IT MATTERS which one you
> pick.
> JUST HOW MANY *further* predicates do you need, to ensure that?
> You went on to say that "this theory is complete", but again, one must
> ask,
> WHICH theory?
[I meant to reply to the group but accidently replied to the poster,
sorry]
you're right, i will try to make the theory more precise and see if i
can clarify what i meant -- however, i don't know how to do it without
invoking = in the theory, so i am hoping you will show me how.
consider the language which has Nat-many constant symbols. now consider
the theory T which says that every c_a != c_b if a != b (where a, b \in
Nat) -- that is, all the constants are distinct from each other. then
it seems like Chris's trick in his post would work in this case to show
that T is not Nat-categorical. that is, given a structure M that
satisfies T, we can make a structure M' that is the same as M except in
its universe we have an extra element, call it k*, such that no
constant names it. then M' will still satisfy T since all of its named
constants are distinct, yet k* will not be mapped to any constant in M,
thus M and M' are non-isomorphic. thus T is not Nat-categorical.
first, is this reasoning correct?
second, why couldn't the same reasoning apply to a cardinal C where C >
Nat? why would such a trick fail then?
and finally: assuming my reasoning is correct and i have applied
Chris's suggestion appropriately, i still cannot find a formula which
is true on M but not true on M', despite having some understanding of
the differences between M and M'. finding such a formula should be
possible since they are not isomorphic. if M is say the model with a
domain like Rat (rational numbers) and we throw in some predicate --
multiplication, addition, or what have you -- how can this be found?
this is quite difficult for me and i was hoping someone could provide
insight as to how to go about it.
appreciate your help very much!
.
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