Coherent Systems of Finite Support Iterations

Abstract

We introduce a forcing technique to construct three-dimensional arrays of generic extensions through FS (finite support) iterations of ccc posets, which we refer to as 3D-coherent systems. We use them to produce models of new constellations in Cichoń's diagram, in particular, a model where the diagram can be separated into 7 different values. Furthermore, we show that this constellation of 7 values is consistent with the existence of a ∆ 1 3 well-order of the reals. IntroductionIn this paper, we provide a generalization of the method of matrix iteration, to which we refer as 3D-coherent systems of iterations and which can be considered a natural extension of the matrix method to include a third dimension. That is, if a matrix iteration can be considered as a system of partial orders P α,β : α ≤ γ, β ≤ δ such that whenever α ≤ α and β ≤ β then P α,β is a complete suborder of P α ,β , then our 3D-coherent systems are systems of posets P α,β,ξ :As an application of this method, we construct models where Cichoń's diagram is separated into different values, one of them with 7 different values. Moreover, these models determine the value of a, which is actually the same as the value of b, and we further show that such models can be produced so that they satisfy, additionally, the existence of a ∆ 1 3 well-order of the reals. The method of matrix iterations, or 2D-coherent systems of iterations in our terminology, has already a long history. It was introduced by Blass and Shelah in [BS89], to show that consistently u < d, where u is the ultrafilter number and d is the dominating number. The method was further developed in [BF11], where the terminology matrix iteration appeared for the first time, to show that if κ < λ are arbitrary regular uncountable cardinals then there is a generic extension in which a = b = κ < s = λ. Here a, b and s denote the almost disjointness, bounding and splitting numbers respectively. In [BF11], the authors also introduce a new method for the preservation of a mad (maximal almost disjoint) family along a matrix iteration, specifically a mad family added by H κ (Hechler's poset for adding a mad family, see Definition 4.1), a method which is of particular importance for our current work. Later, classical preservation properties for matrix iterations were improved by Mejía [Mej13a] to provide several examples of models where the cardinals in Cichoń's diagram assume many different values, in particular, a model with 6 different values. Since then, the question of how many distinct values there can be simultaneously in Cichoń's diagram has been of interest for many authors, see for example [FGKS] (a 2010 Mathematics Subject Classification. 03E17, 03E15, 03E35, 03E40, 03E45.

Matrix Iterations with Vertical Support Restrictions

“…The theory of Brendle and Fischer [BF11] for preserving mad families is the cornerstone for the preservation results we propose in this paper, as well as it is in [FFMM18]. In the latter reference it is proved that E (the standard σ-centered poset adding an eventually different real) and random forcing (thus any random algebra) behaves well in their preservation theory, which allows to prove in [FFMM18,Thm. 4.17] that, whenever κ is an uncountable regular cardinal, the mad family added by the Hechler poset H κ is preserved by any further FS iteration whose iterands are either E, a random algebra or a ccc poset of size < κ.…”

“…Background. In the framework of FS (finite support) iterations of ccc posets to prove consistency results with large continuum (that is, the size of the continuum c = 2 ℵ 0 larger than ℵ 2 ), very recently in [FFMM18] appeared the general notion of coherent systems of FS iterations that was used to construct a three-dimensional array of ccc posets to force that the cardinals in Cichoń's diagram are separated into 7 different values (see Figure 1). This is the first example of a 3D iteration that was used to prove a new consistency result.…”

“…In relation to this, we define a class of posets, which we call σ-Frechet-linked (see Definition 3.24), that includes E and random forcing, and we prove that any (definable) poset in this class behaves well with Brendle's and Fischer's preservation theory (Theorem 3.27). Moreover, by using coherent systems on a power set P(Ω), ⊆ we generalize [FFMM18,Thm. 4.17] by proving that, whenever Ω is uncountable (not necessarily of regular size), the mad family added by H Ω can be preserved by a large class of FS iterations (which includes the Suslin σ-Frechet-linked posets as iterands, see Theorem 4.1).…”

Cichoń’s diagram and localisation cardinals

“…We turn our attention towards related work and open questions. Several recent results [5,[7][8][9][10][11]15,16] have constructed models in which eight or even all ten conceivably different cardinal characteristics in Cichoń's diagram take different values. The constructions involved are all finite support iterations, however, which necessarily means the left side of Cichoń's diagram must be less than or equal to the right side, in particular non(M) ≤ cov(M) (since the cofinality of the iteration length lies between these two cardinal characteristics).…”

On cardinal characteristics of Yorioka ideals

“…Our notation is closer to . The classical preservation theory of Judah‐Shelah and Brendle corresponds to the case of the definition below.…”