A Hanf number for saturation and omission

Suppose t = (T, T1, p) is a triple of two countable theories T ⊆ T1 in vocabularies τ ⊂ τ1 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 essentially equal to the Löwenheim number of second order logic. In Section 4 we make exact computations of these Hanf numbers and note some interesting distinctions between ‘first order ’ and ‘second order quantification’. In particular, we show that if κ is uncountable, h 3 (Lω,ω(Q), κ) = h 3 (Lω1,ω, κ), where h3 is the ‘normal ’ notion of Hanf function (Definition 4.13.) Newelski asked in [New] whether it is possible to calculate the Hanf number of the following property PN. In a sense made precise in Theorem 0.2, we show the answer is no. In accordance with the original question, we focus on countable vocabularies for the first three sections. We deal with extensions to larger vocabularies in Section 4. Definition 0.1 We say M1 | = t where t = (T, T1, p) is a triple of two theories in vocabularies τ ⊂ τ1, respectively, T ⊆ T1 and p is a τ1-type over the empty set if M1the property: There is a model M1 of T1 which omits p, but M1 is saturated is essentially equal to the L¨owenheim number of second order logic. In Section 4 we make exact computations of these Hanf numbers and note some distinctions between ‘first order’ and ‘second order quantification’. In particular, we show that if is uncountable, h3(L!;!(Q); ) = h3(L!1;!; ), where h3 is the ‘normal’ notion of Hanf function (Definition 4.12.)