Transfering saturation, the finite cover property, and stability

by Baldwin and Grossberg and Shelah. [BGSh:570]
J Symbolic Logic, 2000
Saturation is (mu, kappa)-transferable in T if and only if there is an expansion T_1 of T with |T_1|=|T| such that if M is a mu-saturated model of T_1 and |M| >= kappa then the reduct M restriction L(T) is kappa-saturated. We characterize theories which are superstable without f.c.p., or without f.c.p. as, respectively those where saturation is (aleph_0, lambda)-transferable or (kappa (T), lambda)-transferable for all lambda . Further if for some mu >= |T|, 2^mu > mu^+, stability is equivalent to for all mu >= |T|, saturation is (mu,2^mu)-transferable.

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