Transfering Saturation, The Finite Cover Property, and Stability

Saturation is (μ, κ)-transferable in T if and only if there is an expansion T1 of T with |T1| ≤ |T | + א0 such that if M is a μ-saturated model of T1 and |M | ≥ κ then M |L(T ) is κ-saturated. Theorem A. The following are equivalent for a complete theory T . (1) T does not have the finite cover property. (2) ∀λ ≥ |T |+, saturation is (|T |+, λ)-transferable in T . (3) ∃λ > 2|T | such that saturation is (|T |+, λ)-transferable in T . Theorem B. Suppose that there exists a cardinal μ ≥ |T | such that 2μ > μ+. For a complete theory T , the following are equivalent: (1) T is stable. (2) ∀μ ≥ |T |, saturation is (μ+, 2μ)-transferable in T . (3) ∃μ ≥ |T |, saturation is (μ+, μ++)-transferable in T .