We discuss some problems posed by Ciesielski. For example we show that, consistently, d_c is a singular cardinal and e_c<d_c. Next we prove that the Martin Axiom for sigma --centered forcing notions implies that for every function f:R^2 ---> R there are functions g_n,h_n:R ---> R, n< omega, such that f(x,y)= sum_{n=0}^{infty} g_n(x)h_n(y). Finally, we deal with countably continuous functions and we show that in the Cohen model they are exactly the functions f with the property that (for all U in [R]^{aleph_1})(exists U^* in [U]^{aleph_1}) (f restriction U^* is continuous).
The goal of this note is to construct a uniformly antisymmetric function f:R-> R with a bounded countable range. This answers Problem 1(b) of Ciesielski and Larson. (See also list of problems in Thomson and Problem 2(b) from Ciesielski's survey.) A problem of existence of uniformly antisymmetric function f:R-> R with finite range remains open.
We prove that Arhangelskii's problem has a consistent positive answer: if V\models CH, then for some aleph_1-complete aleph_2-c.c. forcing notion P of cardinality aleph_2 we have that P forces ``CH and there is a Lindelof regular topological space of size aleph_2 with clopen basis with every point of pseudo-character aleph_0 (i.e. each singleton is the intersection of countably many open sets)''. Also, we prove the consistency of: CH+ 2^{aleph_1} > \aleph_2 + "there is no space as above with aleph_2 points" (starting with a weakly compact cardinal).
We answer Question~3.2 from Shelah \cite{Sh:666}: Given a maximal almost disjoint (mad) family $\mathcal A$ of size $\aleph_1$, we construct a forcing ${\mathbb Q}(\mathcal A)$ that has Axiom A, is ${}^ωω$-bounding, preserves selective ultrafilters, has the $\aleph_2$-properness isomorphism condition (p.i.c.), and destroys the mad family $\mathcal A$. We develop a new construction technique for partial orders, combining ladder systems for $ω_1$ with trees of normed creatures. Countable support iteration of the new kind of iterands solves Roitman's problem in the case of $d=\aleph_1$ and also s…
Strong reflection principles with the reflection cardinal $\leq\aleph_1$ or $<2^{\aleph_0}$ imply that the size of the continuum is either $\aleph_1$ or $\aleph_2$ or very large. Thus, the stipulation, that a strong reflection principle should hold, seems to support the trichotomy on the possible size of the continuum. In this article, we examine the situation with the reflection principles and related notions of generic large cardinals.
We answer a long-standing open question by proving in ordinary set theory, ZFC, that the Kaplansky test problems have negative answers for aleph_1-separable abelian groups of cardinality aleph_1. In fact, there is an aleph_1-separable abelian group M such that M is isomorphic to M oplus M oplus M but not to M oplus M .
Using forcing with measured creatures we build a universe of set theory in which: (a) every sup-measurable function f:RxR-->R is measurable, and (b) every function f:R-->R is continuous on a non-measurable set. This answers a question of Balcerzak, Ciesielski and Kharazishvili and von Weizsacker's problem (see Fremlin's list of problems).
Problem 5.1 in page 181 of [Fuc15] asks to find the cardinals $λ$ such that there is a universal abelian $p$-group for purity of cardinality $λ$, i.e., an abelian $p$-group $U_λ$ of cardinality $λ$ such that every abelian $p$-group of cardinality $\leq λ$ purely embeds in $U_λ$. In this paper we use ideas from the theory of abstract elementary classes to show: $\textbf{Theorem.}$ Let $p$ be a prime number. If $λ^{\aleph_0}=λ$ or $\forall μ< λ( μ^{\aleph_0} < λ)$, then there is a universal abelian $p$-group for purity of cardinality $λ$. Moreover for $n\geq 2$, there is a universal abelian $p$-…
We investigate the Ramsey theory of continuous pair-colorings on complete, separable metric spaces, and apply the results to the problem of covering a plane by functions. The homogeneity number hm(c) of a pair-coloring c:[X]^2 -> 2 is the number of c-homogeneous subsets of X needed to cover X. We isolate two continuous pair-colorings on the Cantor space 2^omega, c_min and c_max, which satisfy hm(c_min)\le hm(c_max) and prove: 1. For every Polish space X and every continuous pair-coloring c:[X]^2 -> 2 with hm(c) uncountable: hm(c)= hm(c_min) or hm(c)=hm(c_max) 2. There is a model of set theory …
We prove the consistency of irr(prod limits_{i<kappa}B_i/D)< prod limits_{i<kappa}irr(B_i)/D, where D is an ultrafilter on kappa and each B_i is a Boolean Algebra. This solves the last problem of this form from the Monk's list of problems, that is number 35. The solution applies to many other properties, e.g., Souslinity. Next, we get similar results with kappa = aleph_1 (easily we cannot have it for kappa=aleph_0) and Boolean Algebras B_i (i< kappa) of cardinality < beth_{omega_1}.
We prove in ZFC that for mu >= aleph_2 there is a sigma --ideal I on mu and a Boolean sigma --subalgebra B of the family of subsets of mu which includes I such that the natural homomorphism from B onto B/I cannot be lifted.
Absolute model companionship (AMC) is a strict strengthening of model companionship defined as follows: For a theory $T$, $T_{\exists\vee\forall}$ denotes the logical consequences of $T$ which are boolean combinations of universal sentences. $T^*$ is the AMC of $T$ if it is model complete and $T_{\exists\vee\forall}=T^*_{\exists\vee\forall}$. We use AMC to study the continuum problem and to gauge the expressive power of forcing. We show that (a definable version of) $2^{\aleph_0}=\aleph_2$ is the unique solution to the continuum problem which can be in the AMC of a "partial Morleyization" of t…
Einstein Institute of Mathematics, The Hebrew University of Jerusalem（耶路撒冷希伯来大学爱因斯坦数学研究所）；Rutgers University（罗格斯大学）
We prove that no quantifier-free formula in the language of group theory can define the $\aleph_1$-half graph in a Polish group, thus generalising some results from [6]. We then pose some questions on the space of groups of automorphisms of a given Borel complete class, and observe that this space must contain at least one uncountable group. Finally, we prove some results on the structure of the group of automorphisms of a locally finite group: firstly, we prove that it is not the case that every group of automorphisms of a graph of power $λ$ is the group of automorphism of a locally finite gr…
We describe some (countably many) classes K^{n,e} of finite graphs and prove that if lambda^{aleph_0}= lambda then every lambda^+-chromatic graph of cardinal lambda^+ contains, for some n, e, all members of K^{n,e} as subgraphs. On the other hand, it is consistent for every regular infinite cardinal kappa that there is a kappa^+-chromatic graph on kappa^+ that contains finite subgraphs only from K^{n,e} .
It is proved that for each natural number $n$, if $\left| \mathbb{R} \right| = {\aleph}_{n}$, then there is a coloring of ${\left[ \mathbb{R} \right]}^{n+2}$ into ${\aleph}_{0}$ colors that takes all colors on ${\left[ X \right]}^{n+2}$ whenever $X$ is any set of reals which is homeomorphic to $\mathbb{Q}$. This generalizes a theorem of Baumgartner and sheds further light on a problem of Galvin from the 1970s. Our result also complements and contrasts with our earlier result saying that any coloring of ${\left[ \mathbb{R} \right]}^{2}$ into finitely many colors can be reduced to at most $2$ co…
We prove that some fairly basic questions on automata reading infinite words depend on the models of the axiomatic system ZFC. It is known that there are only three possibilities for the cardinality of the complement of an omega-language $L(A)$ accepted by a Büchi 1-counter automaton $A$. We prove the following surprising result: there exists a 1-counter Büchi automaton $A$ such that the cardinality of the complement $L(A)^-$ of the omega-language $L(A)$ is not determined by ZFC: (1). There is a model $V_1$ of ZFC in which $L(A)^-$ is countable. (2). There is a model $V_2$ of ZFC in which $L(A…
We deal with some pcf investigations mostly motivated by abelian group theory problems and deal their applications to test problems (we expect reasonably wide applications). We prove almost always the existence of aleph_omega-free abelian groups with trivial dual, i.e. no non-trivial homomorphisms to the integers. This relies on investigation of pcf; more specifically, for this we prove that "almost always" there are F subseteq lambda^kappa which are quite free and has black boxes. The "almost always" means that there are strong restrictions on cardinal arithmetic if the universe fails this, t…
In this paper I will construct a non-separable hereditarily Lindelof space (L space) without any additional axiomatic assumptions. I will also show that there is a function f from [omega_1]^2 to omega_1 such that if A,B, subsets of omega_1, are uncountable and x omega_1, then there are a < b in A and B respectively with f(a,b) = x. Previously it was unknown whether such a function existed even if omega_1 was replaced by 2. Finally, I will prove that there is no basis for the uncountable regular Hausdorff spaces of cardinality aleph_1. Each of these results gives a strong refutation of a well k…
Contents of this issue: Workshops on SPM themes; Second workshop on Coverings, Selections and Games in Topology (SPM05); Analysis and Descriptive Set Theory Workshop; Descriptive set theory: Effective methods, equivalence relations; Analysis: typical functions, level sets structure; Organizational and additional details; Research announcements; Cardinal restrictions on some homogeneous compacta; Filters: Topological congruence relations on groups; Inverse Limits and Function Algebras; Ultrafilters and partial products of infinite cyclic groups; More on regular reduced products; Consistency of …
