A Hanf number for saturation and omission: the superstable case

by Baldwin and Shelah. [BlSh:992]
Math Logic Quarterly, 2014
Suppose t =(T,T_1,p) is a triple of two theories in vocabularies tau subset tau_1 with cardinality lambda and a tau_1-type p over the empty set. We show the Hanf number for the property: There is a model M_1 of T_1 which omits p, but M_1 restriction tau is saturated is less than beth_{({2^{{(2^{ lambda})}^+}}){}^{^+}} if T is superstable. If T is required only to be stable, the Hanf number is bounded by the Hanf number of L_{(2^lambda)^+, kappa (T)} . We showed in an earlier paper that without the stability restriction the Hanf number is essentially equal to the L{o}wenheim number of second order logic.

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