Using countable support iterations of S-proper posets, we show that the existence of a ∆ 1 3 definable wellorder of the reals is consistent with each of the following: d < c, b < a = s, b < g.

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.

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We prove that if V = L then there is a Π 1 1 maximal orthogonal (i.e. mutually singular) set of measures on Cantor space. This provides a natural counterpoint to the well-known Theorem of Preiss and Rataj [16] that no analytic set of measures can be maximal orthogonal.2000 Mathematics Subject Classification. 03E15.

We study maximal independent families (m.i.f.) in the projective hierarchy. We show that (a) the existence of a Σ 1 2 m.i.f. is equivalent to the existence of a Π 1 1 m.i.f., (b) in the Cohen model, there are no projective maximal independent families, and (c) in the Sacks model, there is a Π 1 1 m.i.f. We also consider a new cardinal invariant related to the question of destroying or preserving maximal independent families. ∀X ∈ [ω] ω ∃a 1 , . . . , a n , b 1 , . . . , b ℓ ∈ I s.t. σ(ā;b) ⊆ * X or σ(ā;b) ∩ X = * ∅.

We study two ideals which are naturally associated to independent families. The first of them, denoted J A , is characterized by a diagonalization property which allows along a cofinal sequence (the order type of which of uncountable cofinality) of stages along a finite support iteration to adjoin a maximal independent family. The second ideal, denoted id(A), originates in Shelah's proof of i < u in Shelah (Arch Math Log 31(6), 433-443, 1992). We show that for every independent family A, id(A) ⊆ J A and define a class of maximal independent families, to which we refer as densely maximal, for which the two ideals coincide. Building upon the techniques of Shelah (1992) we characterize Sacks indestructibility for such families in terms of properties of id(A) and devise a countably closed poset which adjoins a Sacks indestructible densely maximal independent family. Keywords Independent families • Sacks indestructibility • Constellations of cardinal characteristics Mathematics Subject Classification 03E17 • 03E35 The authors would like to thank the Austrian Science Fund (FWF) for the generous support through START Grant Y1012-N35.

We provide a model where u(κ) = κ + < 2 κ for a supercompact cardinal κ.[10] provides a sketch of how to obtain such a model by modifying the construction in [6]. We provide here a complete proof using a different modification of [6] and further study the values of other natural generalizations of classical cardinal characteristics in our model. For this purpose we generalize some standard facts that hold in the countable case as well as some classical forcing notions and their properties.

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.