Model theory for a compact cardinal

by Shelah. [Sh:1019]

We like to develop model theory for T, a complete theory in {L}_{theta, theta}(tau) when theta is a compact cardinal. By [Sh:300a] we have bare bones stability and it seemed we can go no further. Dealing with ultrapowers (and ultraproducts) we restrict ourselves to ``D a theta-complete ultrafilter on I, probably (I, theta)-regular''. The basic theorems work, but can we generalize deeper parts of model theory? In particular can we generalize stability enough to generalize [Sh:c, Ch.VI]? We prove that at least we can characterize the T 's which are minimal under Keisler's order, i.e. such that {D:D is a regular ultrafilter on lambda and M models T => M^lambda /D is lambda-saturated} . Further we succeed to connect our investigation with the logic {L}^1_{< theta} introduced in [Sh:797]: two models are {L}^1_{< theta}-equivalent iff for some omega- sequence of theta-complete ultrafilters, the iterated ultra-powers by it of those two models are isomorphic. 2013.11.14 Doron will read it for a grade, have to finish till 2013.12.31. Updates 2013.12.27 Have update, revise section 2 strongly now have both versions of minimality- different characterization;

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