We prove the following theorem: For a partially ordered set Q such that every countable subset has a strict upper bound, there is a forcing notion satisfying ccc such that, in the forcing model, there is a basis of the null ideal of the real line which is order-isomorphic to Q with respect to set-inclusion. This is a variation of Hechler's classical result in the theory of forcing, and the statement of the theorem for the meager ideal has been already proved by Bartoszynski and the author.
We prove the following theorem: For a partially ordered set Q such that every countable subset has a strict upper bound, there is a forcing notion satisfying ccc such that, in the forcing model, there is a basis of the meager ideal of the real line which is order-isomorphic to Q with respect to set-inclusion. This is a variation of Hechler's classical result in the theory of forcing.
Our main result is that possibly some non-null set of reals cannot be divided to uncountably many non-null sets. We deal also with a non-null set of reals, the graph of any function from it is null and deal with our iterations somewhat more generally.
In this note we answer the following question of Grinblat: Is it consistent that for some set X, cov(NULL restriction X)=lambda is a weakly inaccessible cardinal (so X not null of course) while cov(meagre) is small, say it is aleph_1.
In this paper we introduce a tree-like forcing notion extending some properties of the random forcing in the context of the generalised Cantor space and study its associated ideal of null sets and notion of measurability. This issue was also addressed by Shelah ([11, Problem 0.5]) and concerns the definition of a forcing which is $κ^kappa$-bounding, $< κ$-closed and $κ^+$-cc, for $κ$ inaccessible.
Technical University of Košice（科希策理工大学）；Pavol Jozef Šafárik University in Košice（科希策帕沃尔·约瑟夫·沙法里克大学）；Slovak Academy of Sciences（斯洛伐克科学院）；Technische Universität Wien（维也纳工业大学）；University of Vienna（维也纳大学）
It is well known to generalize the meagre ideal replacing aleph_0 by a (regular) cardinal lambda > aleph_0 and requiring the ideal to be lambda^+-complete. But can we generalize the null ideal? In terms of forcing, this means finding a forcing notion similar to the random real forcing, replacing aleph_0 by lambda, so requiring it to be (<lambda)-complete. Of course, we would welcome additional properties generalizing the ones of the random real forcing. Returning to the ideal (instead forcing) we may look at the Boolean Algebra of lambda-Borel sets modulo the ideal. Surprisingly we get an posi…
With each of the usual tree forcings I (e.g., I = Sacks forcing S, Laver forcing L, Miller forcing M, Mathias forcing R, etc.) we associate a sigma--ideal i^0 on the reals as follows: A \in i^0 iff for all T \in I there is S \leq T (i.e. S is stronger than T or, equivalently, S is a subtree of T) such that A \cap [S] = \emptyset, where [S] denotes the set of branches through S. So, s^0 is the ideal of Marczewski null sets, r^0 is the ideal of Ramsey null sets (nowhere Ramsey sets) etc. We show (in ZFC) that whenever i^0, j^0 are two such ideals, then i^0 \sem j^0 \neq \emptyset. E.g., for I=S …
We show that the set of codes for Ramsey positive analytic sets is $\mathbfΣ^1_2$-complete. This is a one projective-step higher analogue of the Hurewicz theorem saying that the set of codes for uncountable analytic sets is $\mathbfΣ^1_1$-complete. This shows a close resemblance between the Sacks forcing and the Mathias forcing. In particular, we get that the $σ$-ideal of Ramsey null sets is not ZFC-correct. This solves a problem posed by Ikegami, Pawlikowski and Zapletal.
We introduce the notion of effective Axiom A and use it to show that some popular tree forcings are Suslin+. We introduce transitive nep and present a simplified version of Shelah's "preserving a little implies preserving much": If I is a Suslin ccc ideal (e.g. Lebesgue-null or meager) and P is a transitive nep forcing (e.g. P is Suslin+) and P doesn't make any I-positive Borel set small, then P doesn't make any I-positive set small.
J. Zapletal asked if all the forcing notions considered in his monograph are homogeneous. Specifically, he asked if the forcing consisting of Borel sets of $σ$-finite 2-dimensional Hausdorff measure in $\mathbb{R}^3$ (ordered under inclusion) is homogeneous. We give a partial negative answer to both questions by showing that this $σ$-ideal is not homogeneous. Let $\mathcal{N}^1_2$ be the $σ$-ideal of sets in the plane of 1-dimensional Hausdorff measure zero. D. H. Fremlin determined the position of the cardinal invariants of this $σ$-ideal in the Cichoń Diagram. This required proving numerous …
We propose a new, game-theoretic, approach to the idealized forcing, in terms of fusion games. This generalizes the classical approach to the Sacks and the Miller forcing. For definable ($\mathbfΠ^1_1$ on $\mathbfΣ^1_1) $σ$-ideals we show that if a $σ$-ideal is generated by closed sets, then it is generated by closed sets in all forcing extensions. We also prove an infinite-dimensional version of the Solecki dichotomy for analytic sets. Among examples, we investigate the $σ$-ideal $\E$ generated by closed null sets and $σ$-ideals connected with not piecewise continuous functions.
Let ${\bf x}=(x_n)_n$ be a sequence in a Banach space. A set $A\subseteq \mathbb{N}$ is perfectly bounded, if there is $M$ such that $\|\sum_{n\in F}x_n\|\leq M$ for every finite $F\subseteq A$. The collection $B({\bf x})$ of all perfectly bounded sets is an ideal of subsets of $\mathbb{N}$. We show that an ideal $\mathcal{I}$ is of the form $B({\bf x})$ iff there is a non pathological lower semicontinuous submeasure $φ$ on $\mathbb{N}$ such that $\mathcal{I} =FIN(φ)=\{A\subseteq \mathbb{N}: \;φ(A)<\infty\}$. We address the questions of when $FIN(φ)$ is a tall ideal and has a Borel selector. W…
The additivity spectrum ADD(I) of an ideal I is the set of all regular cardinals kappa such that there is an increasing chain {A_alpha:alpha<kappa\} in the ideal I such that the union of the chain is not in I. We investigate which set A of regular cardinals can be the additivity spectrum of certain ideals. Assume that I=B or I=N, where B denotes the sigma-ideal generated by the compact subsets of the Baire space omega^omega, and N is the ideal of the null sets. For countable sets we give a full characterization of the additivity spectrum of I: a non-empty countable set A of uncountable regular…
The $σ$-ideal $(v^0)$ is associated with the Silver forcing, see \cite{bre}. Also, it constitutes the family of all completely doughnut null sets, see \cite{hal}. We introduce segments and $*$-segments topologies, to state some resemblances of $(v^0)$ to the family of Ramsey null sets. To describe $add(v^0)$ we adopt a proof of Base Matrix Lemma. Consistent results are stated, too. Halbeisen's conjecture $cov(v^0) = add(v^0)$ is confirmed under the hypothesis $t= \min \{\cf (\frak c), r\} $. The hypothesis $h=ω_1$ implies that $(v^0)$ has the ideal type $(\frak c, ω_1,\frak c)$.
I prove forcing preservation theorems for products of definable partial orders preserving the cofinality of the meager or null ideal. Rectangular Ramsey theorems for related ideals follow from the proofs.
Let $G$ be an abelian Polish group, e.g. a separable Banach space. A subset $X \subset G$ is called Haar null (in the sense of Christensen) if there exists a Borel set $B \supset X$ and a Borel probability measure $μ$ on $G$ such that $μ(B+g)=0$ for every $g \in G$. The term shy is also commonly used for Haar null, and co-Haar null sets are often called prevalent. Answering an old question of Mycielski we show that if $G$ is not locally compact then there exists a Borel Haar null set that is not contained in any $G_δ$ Haar null set. We also show that $G_δ$ can be replaced by any other class of…
A subset $X$ of a Polish group $G$ is \emph{Haar null} if there exists a Borel probability measure $μ$ and a Borel set $B$ containing $X$ such that $μ(gBh)=0$ for every $g,h \in G$. A set $X$ is \emph{Haar meager} if there exists a compact metric space $K$, a continuous function $f : K \to G$ and a Borel set $B$ containing $X$ such that $f^{-1}(gBh)$ is meager in $K$ for every $g,h \in G$. We calculate (in $ZFC$) the four cardinal invariants ($\rm add$, $\rm cov$, $\rm non$, $\rm cof$) of these two $σ$-ideals for the simplest non-locally compact Polish group, namely in the case $G = \mathbb{Z}…
