[FOM] Question about Conservative Extensions
Craig Smorynski
smorynski at sbcglobal.net
Thu Sep 13 17:05:36 EDT 2012
Trivial examples are easy to construct. For example, let T1 be the (complete) theory of successor, Th(N,0,S) and T2 be Presburger-Skolem arithmetic. T2 is conservative over T1 because (N,0,S) extends to a model of T2. But the structure consisting of N and a copy of the integers with their respective successor functions cannot be extended to include addition.
On Sep 13, 2012, at 11:10 AM, 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?
>
> 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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Craig
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