THE CONSISTENCY OF ARBITRARILY LARGE SPREAD BETWEEN u AND d

As an application of the method used to obtain the result above, we will obtain the consistency of b = ω1 < s = κ where b is the bounding number, s is the splitting number and κ is an arbitrary regular cardinal. Suppose ν ≥ δ. Begin with a model of GCH and adjoin δ-many Cohen reals 〈rα : α ∈ δ〉 followed by ν-many Random reals 〈sξ : ξ ∈ ν〉. That is, if Vδ is the model obtained after the first δ Cohen reals, the generic extension in which we are interested is obtained by finite support iteration of length ν of Random real forcing over Vδ. Since random forcing is ω-bounding, the Cohen reals remain a dominating family in the final generic extension Vδ,ν . Furthermore for any family of reals of size smaller than δ there is a Cohen real which is unbounded by this family, and so Vδ,ν d = δ. To verify that u = ν, recall that if a is random real over some model M , then neither a, no ω−a contains infinite sets from M . Again since the ground model V satisfies GCH and the forcing notions with which we work have the countable chain condition, any set of reals A in Vδ,ν of size smaller than ν is obtained at some initial stage of the random real forcing iteration Vδ,α for some α < ν. But then neither sα nor ω − sα contains an element of A and so A does not generate an ultrafilter. Therefore u = c = ν.