[FOM] Question about Conservative Extensions

[John Baldwin]

On Thu, 13 Sep 2012, Richard Heck 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?
>

Let T be the complete theory which says that E_1 is an equivalence
relation which has infinitely many infinite extensions. Since it is
complete, any extension is conservative.  Let T_1 be the extension in a 
language with one further binary relation E_2 that asserts that E_2 is 
also an equivalenc relation and that each E_2 
class intersects each E_1 class in exactly one element.

The only models of T_1 that can be expanded to models of T_2 are those 
where every class has the same size.

John T. Baldwin
Professor Emeritus
Department of Mathematics, Statistics,
and Computer Science M/C 249
jbaldwin at uic.edu
851 S. Morgan
Chicago IL
60607





> Thanks,
> Richard Heck
>
> -- 
> -----------------------
> Richard G Heck Jr
> Romeo Elton Professor of Natural Theology
> Brown University
>
> Check out my book Frege's Theorem:
>  http://tinyurl.com/fregestheorem
> Visit my website:
>  http://frege.brown.edu/heck/
>
>
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