Dependent $T$ and Existence of limit models

by Shelah. [Sh:877]

We continue [Sh:868] and [Sh:783]. 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^{< lambda}> kappa = cf (kappa)>|T| . The stable theories has one; are there unstable T whnce of limit models AUTHORS: Saharon Shelah ich 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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