[FOM] Question about Conservative Extensions
Noah David Schweber
schweber at math.berkeley.edu
Thu Sep 13 22:57:23 EDT 2012
I might be terribly confused, but isn't the following a simple
counterexample?
Take T1 to be the theory - with signature consisting of two unary
predicates U and V - asserting that U and V partition the universe into two
disjoint infinite sets.
Now consider the expanded language with an added unary function symbol, f,
and the theory in this expanded language T2 consisting of T1 + "f is a
bijection from U to V."
T2 is a conservative extension of T1; but if we consider a model of T1 in
which U is countable and V is uncountable, then this model cannot be
extended to a model of T2.
Does this work?
- Noah
On Thu, 13 Sep 2012 12:10:58 -0400, Richard Heck <rgheck at brown.edu> wrote:
> Hi, all,
>
> I was introducing my students today to model-theoretic proofs that some
> theory is a conservative extension of another, and one of them asked me,
> in effect, when the converse of the usual argument is also true, i.e.:
> If T2 is a conservative extension of T1, can every model of T1 always be
> expanded to a model of T2? I believe the answer must be "no", and that
> models of PA that do not have satisfaction classes would provide one
> counter-example. Is that right? If not, are there other examples? And
> even if so, are there (much) simpler examples?
>
> Thanks,
> Richard Heck
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