4
$\begingroup$

Let $M$ be a countable structure that is homogeneous, i.e. every isomorphism between finitely generated substructures of $M$ extends to an automorphism of $M$. Let $\phi_M$ be the Scott sentence of $M$.

If $N$ is an uncountable model of $\phi_M$, then $N$ is not necessarily homogeneous, but is the existence of an uncountable model sufficient for the existence of an uncountable homogeneous model? If so, can we further conclude that given $N\models\phi_M$, $N$ uncountable, there is a homogenous model of $\phi_M$ of the same cardinality as $N$?

Any help?

$\endgroup$
3
  • $\begingroup$ @Ioannis: there is an interesting special case of the question you are asking, when M is a countable recursively saturated model of $PA$, for which I suspect that the answer to your first question is negative. If I make any progress, I will let you know. $\endgroup$
    – Ali Enayat
    Jun 19, 2011 at 17:41
  • 1
    $\begingroup$ @Ioannis: Update, it turns out that the answer to your question is POSITIVE for the recursively saturated $M$; which you may have already known about, if not, see Theorem 10 of: S. Buechler, Steven, Expansions of models of $\omega $ω-stable theories. J. Symbolic Logic 49 (1984), no. 2, 470–477 $\endgroup$
    – Ali Enayat
    Jun 21, 2011 at 14:24
  • $\begingroup$ Thank you for the reference. I will have to go through it to clarify all the details. It seems to me a natural question to ask. Someone might have thought about it already even in the non-recursive case. $\endgroup$ Jun 21, 2011 at 16:12

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.