The work presented is joint with S-D Friedman and T Hyttinen. We aim to generalize a very nice result of Friedman, Hyttinen, and Rautila, which ties first-order model theoretic classification theory to constructibility under the assumption of \(0^\sharp\), to a non-elementary model theoretic setting. The original result stated:
Theorem. Assume \(0^\sharp\) exists and let \(T\) be a constructible first-order theory which is countable in the constructible universe \(L\). Let \(\kappa\) be a cardinal in \(L\) larger than \((\aleph_1)^L\). Then the collection of constructible pairs of models \(A,B\) of \(T\), \(|A|,|B|=\kappa\), which are isomorphic in a cardinal- and real-preserving extension of \(L\) is itself constructible if and only if \(T\) is classifiable (i.e. superstable with NDOP and NOTOP).
We have chosen Homogeneous Model Theory as a good setting for generalizing this result.
In Part I of this talk, a gentle introduction to Homogeneous Model Theory was given, as well as a justification as to why this is a good setting to choose.
In Part II, one easy step for our generalization will be sketched: the unstable case.