Jörg Brendle

Solovay-type characterizations for forcing algebras

We give characterizations for the (in ZFC unprovable) sentences “Every -set is measurable” and “Every -set is measurable” for various notions of measurability derived from well-known forcing partial orderings.

Forcing indestructibility of MAD families

Abstract Let A ⊆ [ ω ] ω be a maximal almost disjoint family and assume P is a forcing notion. Say A is P -indestructible if A is still maximal in any P -generic extension. We investigate P -indestructibility for several classical forcing notions P . In particular, we provide a combinatorial characterization of P -indestructibility and, assuming a fragment of MA, we construct maximal almost disjoint families which are P -indestructible yet Q -destructible for several pairs of forcing notions ( P , Q ) . We close with a detailed investigation of iterated Sacks indestructibility.

Silver measurability and its relation to other regularity properties

For a ⊆ b ⊆ ω with b\a infinite, the set D = {x ∈ [ω] : a ⊆ x ⊆ b} is called a doughnut. Doughnuts are equivalent to conditions of Silver forcing, and so, a set S ⊆ [ω] is called Silver measurable, also known as completely doughnut, if for every doughnut D there is a doughnut D′ ⊆ D which is contained or disjoint from S. In this paper, we investigate the Silver measurability of ∆2 and Σ 1 2 sets of reals and compare it to other regularity properties like the Baire and the Ramsey property and Miller and Sacks measurability.

The almost-disjointness number may have countable cofinality

We show that it is consistent for the almost-disjointness number a to have countable cofinality. For example, it may be equal to N ω .

Regularity properties for dominating projective sets

Abstract We show that every dominating analytic set in the Baire space has a dominating closed subset. This improves a theorem of Spinas [15] saying that every dominating analytic set contains the branches of a uniform tree, i.e. a superperfect tree with the property that for every splitnode all the successor splitnodes have the same length. In [15], a subset of the Baire space is called u-regular if either it is not dominating or it contains the branches of a uniform tree, and it was proved that Σ 2 1 - K σ -regularity implies Σ 2 1 - u -regularity. Here we show that these properties are in fact equivalent. Since the proof of analytic u -regularity uses a game argument it was clear that (projective) determinacy implies u -regularity of all (projective) sets. Here we show that an inaccessible cardinal is enough to construct a model for projective u -regularity, namely it holds in Solovays model. Finally we show that forcing with uniform trees is equivalent to Laver forcing.

Cardinal invariants of the continuum and combinatorics on uncountable cardinals

Abstract We explore the connection between combinatorial principles on uncountable cardinals, like stick and club, on the one hand, and the combinatorics of sets of reals and, in particular, cardinal invariants of the continuum, on the other hand. For example, we prove that additivity of measure implies that Martin’s axiom holds for any Cohen algebra. We construct a model in which club holds, yet the covering number of the null ideal cov ( N ) is large. We show that for uncountable cardinals κ ≤ λ and F ⊆ [ λ ] κ , if all subsets of λ either contain, or are disjoint from, a member of F , then F has size at least cov ( N ) etc. As an application, we solve the Gross space problem under c = ℵ 2 by showing that there is such a space over any countable field. In two appendices, we solve problems of Fuchino, Shelah and Soukup, and of Kraszewski, respectively.

Martin's Axiom and the Dual Distributivity Number

We show that it is consistent that Martins axiom holds, the continuum is large, and yet the dual distributivity number ℌ is κ1. This answers a question of Halbeisen.

Evasion and prediction

Abstract. Say that a function π:n<ω→n (henceforth called a predictor) k-constantly predicts a real xnω if for almost all intervals I of length k, there is iI such that x(i)=π(x↾i). We study the k-constant prediction number vnconst(k), that is, the size of the least family of predictors needed to k-constantly predict all reals, for different values of n and k, and investigate their relationship.

Mutual Generics and Perfect Free Subsets

We present two ways of adjoining a perfect set of mutually random reals to a model V of ZFC. We show that adding a random real does not add a perfect set of mutually non-constructible reals. We also investigate the existence of perfect free subsets for projective functions f : (ww)n → ww.

Bounding, splitting, and almost disjointness

Abstract We investigate some aspects of bounding, splitting, and almost disjointness. In particular, we investigate the relationship between the bounding number, the closed almost disjointness number, the splitting number, and the existence of certain kinds of splitting families.