Mad families, splitting families and large continuum

Abstract

Abstract. Let κ < λ be regular uncountable cardinals. Using a finite support iteration of ccc posets we obtain the consistency of b = a = κ < s = λ. If µ is a measurable cardinal and µ < κ < λ, then using similar techniques we obtain the consistency of b = κ < a = s = λ.

Matrix iterations and Cichon’s diagram

“…In the same general context presented in section 2, preservation results about unbounded reals are contained in section 4. Those results have been already presented in [3] and [5] with a particular notation, but we add Theorem 7 concerning these preservation results with the properties stated in section 2. In section 5, we define the specific case of matrix iterations of ccc posets and present Corollary 1 that allows us to calculate, in a generic extension, the size of one invariant of the right hand side of Cichon's diagram.…”

“…The last two definitions are important notions introduced in [3] and [5] for the preservation of unbounded reals and the construction of matrix iterations. In relation with the preservation property of Definition 2, we have the following.…”

“…Theorem 8 (Blass and Shelah, [3], [5,Lemma 12]). With the notation in Lemma 11, assume that P 0,δ M P 1,δ .…”

“…Nevertheless, models where invariants of the right hand side of Cichon's diagram can assume more than two arbitrary values seem more difficult to get, furthermore, more sophisticated techniques than the usual finite support iteration of ccc posets seem to be needed to construct such models. We use the technique of matrix iterations of ccc posets (see [3] and [5]) to construct models of ZFC of some cases where cardinals on the right hand side of Cichon's diagram can assume two or three arbitrary values, the greatest of them with uncountable cofinality and the others regular. Even more, we use some of the reasonings in [4] with this technique in order to, at the same time, assign several arbitrary regular values to the invariants of the left hand side of the diagram.…”

On the cofinality of the splitting number

“…By the induction hypothesis, the family {x i : i ∈ I} is θ-Luzin in V [G β ] and so we have that {x i : i ∈ J 3 } is finite and {x i : i ∈ J 3 } is co-finite. Choose any q < p in G β and a nameJ 3 for J 3 so that q forces this property forJ 3 . Since q forces thatJ 3 ⊂J 0 , we have that q forces the same property forJ 0 .…”