Logic Seminar
John Goodrick
University of Maryland
Bi-embeddability and isomorphism: the weakly minimal case.
Abstract: Recently (in joint work with Chris Laskowski) we characterized countable, weakly minimal theories $T$ such that any two (elementarily) bi-embeddable models of $T$ are isomorphic. We prove that
if $T$ is countable and weakly minimal, the following are equivalent:
1. $T$ has two bi-embeddable but nonisomorphic models;
2. There is an automorphism $f$ of the monster model of $T$ and a strong type $p$ over the empty set which is almost-orthogonal to $f(p) \otimes \ldots \otimes f^n(p) $ for any n.
3. $T$ has an infinite collection of models that are pairwise bi-embeddable but pairwise nonisomorphic.
The proof involves some geometric stability theory plus a Dushnik-Miller type argument to build nonisomorphic models by "killing'' every potential isomorphism at each stage of the construction.
seminar begins with tea
Tuesday November 3, 2009 at 4:00 PM in SEO 612