Dependent T and existence of limit models

Details Dependent T and existence of limit models Dependent T and existence of limit models

2006-09-22

arxiv.org/abs/math/0609636v2

Mathematics

Logic

Scientific paper

We continue math.LO/0603651 and math.LO/0406440. The problem there is when does (first order) T have a model M of cardinality lambda which is (one of the variants of) a limit model for cofinality kappa, and the most natural case to try is lambda = lambda^{ kappa =cf(kappa)>|T|. The stable theories has one; are there unstable T which has such limit models? We find one: the theory T_{ord} of dense linear orders. So does this hold for all unstable T ? As T_{ord} is prototypical of dependent theories, it is natural to look for independent theories. A strong, explicit version of T being independent is having the strong independence property. We prove that for such T there are no limit models. We work harder to prove this for every dependent T, i.e., with the independence property though a weaker version. This makes us conjecture that any dependent T has such models. Toward this end we continue the investigation of types for dependent T.

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