Model Theory for theta-Complete Ultrapowers
Saharon Shelah · arXiv (Cornell University) · 2013
We would like to develop model theory for T, a complete theory in L_{theta,theta}(tau) when theta is a compact cardinal. We have bare bones stability theory and it seemed we can go no further. Dealing with ultrapowers (and ultraproducts) naturally we restrict ourselves to D a theta-complete ultrafilter on I, probably (I,theta)-regular. The basic theorems work and can be generalized (like Los theorem), but can we generalize deeper parts of model theory? In particular, can we generalize stability enough to generalize [Sh:c, Ch. VI]? Let us concentrate on saturated in the local sense (types consisting of instances of one formula). We prove that at least we can characterize the T's (of cardinality 2^kappa +|T| in each of the following two senses. One is generalizing Keisler order which measures how saturated are ultrapowers. Another asks: Is there an L_{theta,theta}-theory T_1 supseteq T of cardinality |T| + 2^theta such that for every model M_1 of T_1 of cardinality > lambda, the tau(T)-reduct M of M_1 is lambda^+-saturated. Moreover, the two versions of stable used in the characterization are different. Further we succeed to connect our investigation with the logic L^1_{< theta} introduced in [Sh:797] proving it satisfies several parallel of classical theorems on first order logic, strengthening the thesis that it is a natural logic. In particular, 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.