2 research outputs found
Model Theory for a Compact Cardinal
We would like to develop model theory for T, a complete theory in
L_{theta,theta}(tau) when theta is a compact cardinal. We already 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 of model theory work and can be generalized (like Los theorem), but
can we generalize deeper parts of model theory? The first section tries to sort
out what occurs to the notion of stable T for complete L_{theta,theta}-theories
T. We generalize several properties of complete first order T, equivalent to
being stable (see [Sh:c]) and find out which implications hold and which fail.
In particular, can we generalize stability enough to generalize [Sh:c, Ch. VI]?
Let us concentrate on saturation in the local sense (types consisting of
instances of one formula). We prove that at least we can characterize the T's
(of cardinality < theta for simplicity) which are minimal for appropriate
cardinal lambda > 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