### 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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