A Hanf number for saturation and omission: the superstable case

Suppose t = (T, T1, p) is a triple of two theories in vocabularies τ ⊂ τ1 with cardinality λ and a τ1-type p over the empty set. We show the Hanf number for the property: ‘There is a model M1 of T1 which omits p, but M1 τ is saturated’ is less than i (2(2 λ)+ ) + if T is superstable. We showed in [BS11] that with no stability restriction the Hanf number is essentially equal to the Löwenheim number of second order logic. Hanf observed [Han60] that if one asks for each K in a set of classes of structures, ‘Does K have arbitrarily large members?’, there is a cardinal κ (the sup of the maxima of the bounded K) such that any class with a member at least of cardinality κ has arbitrarily large models. In many cases this bound κ can be calculated (For a countable first order theory, it is א0.) In this paper we call a Hanf number for a family K of classes calculable if it is bounded by a function that can be computed by an arithmetic function in ZFC (See Definition 0.1.) and if not it is incalculable. The following definition is more abstract than needed for this paper but we include it for comparison with other works where other Hanf functions are shown to be not calculable. Definition 0.1 1. A function f (a class-function from cardinals to cardinals) is strongly calculable if f can (provably in ZFC) be defined in terms of cardinal addition, multiplication, exponentiation, and iteration of the i function. ∗Baldwin was partially supported by NSF-0500841 and he wishes to thank the John Templeton Foundation for its support through Project #13152, Myriad Aspects of Infinity, hosted during 2009-2011 at the Centre de Recerca Matematica, Bellaterra, Spain. Both thank the NSF and Rutgers University for support of the visit where this work was done. For partial support of this research Shelah thanks the Israel Science Foundation (Grant no. 1053/11), and the National Science Foundation (Grant No.DMS 1101597). This paper is BlSh 992 on the Shelah Archive.