Hanf number for the strictly stable cases

TL;DR

This paper determines the Hanf number for models omitting a type and saturated on a sub-vocabulary in the context of stable theories, establishing bounds relative to known Hanf numbers in infinitary logic.

Contribution

It characterizes the Hanf number for the property of models omitting a type and being saturated in the stable case, with specific bounds related to infinitary logic Hanf numbers.

Findings

Hanf number exceeds that of L_{lambda^+, kappa}

Hanf number is less than that of L_{(2^lambda)^+, kappa}

Provides a precise characterization for stable theories with fixed parameters

Abstract

Suppose t = (T,T_1, p) is a triple of two theories T subset T_1 in vocabularies tau subset tau_1 (respectively) of cardinality lambda and a tau_1-type p over the empty set; in the main case here is with T stable. We show the Hanf number for the property: "there is a model M_1 of T_1 which omits p, but M_1 restricted to tau is saturated" is larger than the Hanf number of L_{lambda^+, kappa} but smaller than the Hanf number of L_{(2^lambda)^+, kappa} when T is stable with kappa = kappa(T). In fact, we characterize the Hanf number of t when we fix (T, lambda) where T is a first order complete, lambda > |T| and demand |T_1| < lambda.