On the cofinality of the splitting number

The splitting number $\mathfrak{s}$ can be singular. The key method is to construct a forcing poset with finite support matrix iterations of ccc posets introduced by Blass and Shelah (1989).     

Groupwise density cannot be much bigger than the unbounded number

We prove that g (the groupwise density number) is smaller or equal to ??⁺, the successor of the minimal cardinality of an unbounded subset of ^ωω. This is true even for the version of ?? for groupwise dense ideals. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Bounding,splitting, and almost disjointness

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.     

Remarks on small sets of reals

We show that the Dual Borel Conjecture implies that \boldsymbol\aleph_1 $"> and find some topological characterizations of perfectly meager and universally meager sets.

     

On Shattering,Splitting and Reaping Partitions

In this article we investigate the dual-shattering cardinal ?, the dual-splitting cardinal ?? and the dual-reaping cardinal ??, which are dualizations of the well-known cardinals ?? (the shattering cardinal, also known as the distributivity number of P(ω)/fin), s (the splitting number) and ?? (the reaping number). Using some properties of the ideal ?? of nowhere dual-Ramsey sets, which is an ideal over the set of partitions of ω, we show that add(??) = cov(??) = ?. With this result we can show that ? > ω₁ is consistent with ZFC and as a corollary we get the relative consistency of ? > ?? t, where t is the tower number. Concerning ?? we show that cov(M) ? ?? ?? (where M is the ideal of the meager sets). For the dual-reaping cardinal ?? we get p ?? ? ?? ? ?? (where ?? is the pseudo-intersection number) and for a modified dual-reaping number ??′ we get ??′ ? ?? (where ?? is the dominating number). As a consistency result we get ?? < cov(??).     

The ring of Fermat reals

We give the definition of the ring of Fermat reals, a simple extension of the real field containing nilpotent infinitesimals. The construction takes inspiration from smooth infinitesimal analysis, but provides a powerful theory of actual infinitesimals without any need of a background in mathematical logic. In particular it is consistent with classical logic. We face the problem to decide if the product of powers of nilpotent infinitesimals is zero or not, the identity principle for polynomials, the characterization of invertible elements and some applications to Taylor's formulas.     

The number of translates of a closed nowhere dense set required to cover a Polish group

For a Polish group let be the minimal number of translates of a fixed closed nowhere dense subset of required to cover . For many locally compact this cardinal is known to be consistently larger than which is the smallest cardinality of a covering of the real line by meagre sets. It is shown that for several non-locally compact groups . For example the equality holds for the group of permutations of the integers, the additive group of a separable Banach space with an unconditional basis and the group of homeomorphisms of various compact spaces.     

A transfinite hierarchy of reals

We extend the hierarchy defined in [5] to cover all hyperarithmetical reals. An intuitive idea is used or the definition, but a characterization of the related classes is obtained. A hierarchy theorem and two fixed point theorems (concerning computations related to the hierarchy) are presented.     

Fermat–Reyes method in the ring of Fermat reals

To discover derivatives, Pierre de Fermat used to assume a non-zero increment h in the incremental ratio and, after some calculations, to set h=0 in the final result. This method, which sounds as inconsistent, can be perfectly formalized with the Fermat–Reyes theorem about existence and uniqueness of a smooth incremental ratio. In the present work, we will introduce the cartesian closed category where to study and prove this theorem and describe in general the Fermat method. The framework is the theory of Fermat reals, an extension of the real field containing nilpotent infinitesimals which does not need any knowledge of mathematical logic. This key theorem will be essential in the development of differential and integral calculus for smooth functions defined on the ring of Fermat reals and also for infinite-dimensional operators like derivatives and integrals.     

CCC forcing and splitting reals

The results of this paper were motivated by a problem of Prikry who asked if it is relatively consistent with the usual axioms of set theory that every nontrivial ccc forcing adds a Cohen or a random real. A natural dividing line is into weakly distributive posets and those which add an unbounded real. In this paper I show that it is relatively consistent that every nonatomic weakly distributive ccc complete Boolean algebra is a Maharam algebra, i.e. carries a continuous strictly positive submeasure. This is deduced from theP-ideal dichotomy, a statement which was first formulated by Abraham and Todorcevic [AT] and later extended by Todorcevic [T]. As an immediate consequence of this and the proof of the consistency of theP-ideal dichotomy we obtain a ZFC result which says that every absolutely ccc weakly distributive complete Boolean algebra is a Maharam algebra. Using a previous theorem of Shelah [Sh1] it also follows that a modified Prikry conjecture holds in the context of Souslin forcing notions, i.e. every nonatomic ccc Souslin forcing either adds a Cohen real or its regular open algebra is a Maharam algebra. Finally, I also show that every nonatomic Maharam algebra adds a splitting real, i.e. a set of integers which neither contains nor is disjoint from an infinite set of integers in the ground model. It follows from the result of [AT] that it is consistent relative to the consistency of ZFC alone that every nonatomic weakly distributive ccc forcing adds a splitting real.     

Maharam algebras and Cohen reals

We show that the product of any two nonatomic Maharam algebras adds a Cohen real. As a corollary of this and a result of Shelah (1994) we obtain that the product of any two nonatomic ccc Souslin forcing notions adds a Cohen real.

     

Projective well orders and coanalytic witnesses

We further develop a forcing notion known as Coding with Perfect Trees and show that this poset preserves, in a strong sense, definable P-points, definable tight MAD families and definable selective independent families. As a result, we obtain a model in which $\mathfrak{a}=\mathfrak{u}=\mathfrak{i}={\mathrm{?}}_1<2^{{\mathrm{?}}_0}={\mathrm{?}}_2$, each of $\mathfrak{a}$, $\mathfrak{u}$, $\mathfrak{i}$ has a ${\mathrm{\Pi }}_1^1$ witness and there is a ${\mathrm{\Delta }}_3^1$ well-order of the reals. Note that both the complexity of the witnesses of the above combinatorial cardinal characteristics, as well as the complexity of the well-order are optimal. In addition, we show that the existence of a ${\mathrm{\Delta }}_3^1$ well-order of the reals is consistent with $\mathfrak{c}={\mathrm{?}}_2$ and each of the following: $\mathfrak{a}=\mathfrak{u}<\mathfrak{i}$, $\mathfrak{a}=\mathfrak{i}<\mathfrak{u}$, $\mathfrak{a}<\mathfrak{u}=\mathfrak{i}$, where the smaller cardinal characteristics have co-analytic witnesses.Our methods allow the preservation of only sufficiently definable witnesses, which significantly differs from other preservation results of this type.     

More on trees and Cohen reals

In this paper we analyse some questions concerning trees on κ, both for the countable and the uncountable case, and the connections with Cohen reals. In particular, we provide a proof for one of the implications left open in [6, Question 5.2] about the diagram for regularity properties.     

Turing cones and set theory of the reals

We investigate Turing cones as sets of reals, and look at the relationship between Turing cones, measures, Baire category and special sets of reals, using these methods to show that Martin's proof of Turing Determinacy (every determined Turing closed set contains a Turing cone or is disjoint from one) does not work when you replace “determined” with “Blackwell determined”. This answers a question of Tony Martin. Received: 6 December 1999 / Revised version: 28 June 2000 Published online: 3 October 2001