Submission date: 4 December 2009.

Abstract:

A theory is stable up to Δ if any Δ-type over a model has a few extensions up to complete types. I prove that a theory has no the independence property iff it is stable up to some Δ, where each φ(x;\bar y) \in Δ has no the independence property. Definability of one-types over a model of a stable up to Δ theory is investigated.

Mathematics Subject Classification: 03C07, 03C45, 03C64

Keywords and phrases: stability, the independence property, dependent theories, definability of types

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