Monk asks (problems 13, 15 in his list; pi is the algebraic density):''For a Boolean algebra B, aleph_0 <= theta <= pi (B), does B have a subalgebra B' with pi (B')= theta ?'' If theta is regular the answer is easily positive, we show that in general it may be negative, but for quite many singular cardinals - it is positive; the theorems are quite complementary. Next we deal with pi-chi and we show that the pi-chi of an ultraproduct of Boolean algebras is not necessarily the ultraproduct of the pi-chi 's. We also prove that for infinite Boolean algebras A_i (i< kappa) and a non-principal ultrafilter D on kappa : if n_i< aleph_0 for i< kappa and mu = prod_{i< kappa} n_i/D is regular, then pi-chi(A) >= mu. Here A= prod_{i< kappa}A_i/D. By a theorem of Peterson the regularity of mu is needed.
在[Sh：89]中，我们回答了蒙克的一个问题，阐明了“除了由超滤子引起的自同态之外没有自同态的布尔代数”的概念（见第2节），并证明了一个具有特征密度aleph_0的布尔代数的存在，首先假设diamondsuit_{aleph_1}，然后只假设CH。其思想是，如果h是B的自同态，而不是“平凡的”自同态，那么B中有成对不相交的D_n，h（d_n）不是子集d_n。然后，我们可以对某个S子集Ω，添加一个元素x，使得对于S中的n，d <= x，对于不在S中的n，x cap d_n=0，同时禁止{y cap h（d_n）：n in S} cup {y cap h（d_n）=0：n not in S}的解。进一步的分析表明，关键是我们忽略了正量词自由类型。继续这蒙克成功地证明了ZFC，存在这样的布尔代数基数2^{aleph_0}。我们证明（在ZFC）存在这样的B的密度字符λ和基数λ ^{aleph_0}时，λ> aleph_0。我们可以从蒙克的清单中总结出一些问题的答案。我们使用[Sh：45]，[Sh：172]中的组合方法，该方法在第1节中表示。
Assuming the existence of a Mahlo cardinal, we produce a generic extension of Gödel's constructible universe $L$, in which the transfer principles $(\aleph_2, \aleph_0) \to (\aleph_3, \aleph_1)$ and $(\aleph_3, \aleph_1) \to (\aleph_2, \aleph_0)$ fail simultaneously. The result answers a question of Silver from 1971. We also extend our result to higher gaps.
We show that if $cf(2^{\aleph_0})=\aleph_1,$ then any non-trivial $\aleph_1$-closed forcing notion of size $\leq 2^{\aleph_0}$ is forcing equivalent to $Add(\aleph_1, 1),$ the Cohen forcing for adding a new Cohen subset of $ω_1.$ We also produce, relative to the existence of suitable large cardinals, a model of $ZFC$ in which $2^{\aleph_0}=\aleph_2$ and all $\aleph_1$-closed forcing notion of size $\leq 2^{\aleph_0}$ collapse $\aleph_2,$ and hence are forcing equivalent to $Add(\aleph_1, 1).$ These results answer a question of Scott Williams from 1978. We also extend a result of Todorcevic and…
The Noetherian type of a space X, Nt(X), is the least cardinal kappa such that X has a base B such that every element of the base is contained in less than kappa many elements of the base. Denote X the space obtained from 2^{aleph_omega} by declaring the G_delta sets to be open. Milovich proved that if Square_{aleph_omega} holds and (aleph_omega)^omega=aleph_{omega+1} then Nt(X)=omega_1. Answering a question of Spadaro, we show that if (aleph_omega)^omega=aleph_{omega+1} and a strong form of Chang Conjecture holds for aleph_ωthen Nt(X)>omega_1.
In this paper, we address the question of when a non-free $\aleph_1$-free group $H$ can be be free in a transitive cardinality-preserving model extension. Using the $Γ$-invariant, denoted $Γ(H)$, we present a necessary and sufficient condition resolving this question for $\aleph_1$-free groups of cardinality $\aleph_1$. Specifically, if $Γ(H) = [\aleph_1]$, then $H$ will be free in a transitive model extension if and only if $\aleph_1$ collapses, while for $Γ(H) \ne [\aleph_1]$ there exist cardinality-preserving forcings that will add a basis to $H$. In particular, for $Γ(H) \neq [\aleph_1]$, …
In [FHK13], the authors considered the question whether model-existence of $L_{ω_1,ω}$-sentences is absolute for transitive models of ZFC, in the sense that if $V \subseteq W$ are transitive models of ZFC with the same ordinals, $φ\in V$ and $V\models "φ\text{ is an } L_{ω_1,ω}\text{-sentence}"$, then $V \models "φ\text{ has a model of size } \aleph_α"$ if and only if $W \models "φ\text{ has a model of size } \aleph_α"$. From [FHK13] we know that the answer is positive for $α=0,1$ and under the negation of CH, the answer is negative for all $α>1$. Under GCH, and assuming the consistency of a s…
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…
Motivated by a question from a recent paper by Gilton, Levine and Stejskalova, we obtain a new characterization of the ideal $J[κ]$, from which we confirm that $κ$-Souslin trees exist in various models of interest. As a corollary we get that for every integer $n$ such that $\mathfrak b<2^{\aleph_n}=\aleph_{n+1}$, if $\square(\aleph_{n+1})$ holds, then there exists an $\aleph_{n+1}$-Souslin tree.
This paper is part II of a study on cardinals that are characterizable by a Scott sentence, continuing the work from http://arxiv.org/abs/1007.2426v1. A cardinal $κ$ is characterized by a Scott sentence $ϕ_M$, if $ϕ_M$ has a model of size $κ$, but no model of $κ^+$. The main question in this paper is the following: Are the characterizable cardinals closed under the powerset operation? We prove that if $\aleph_β$ is characterized by a Scott sentence, then $2^{\aleph_{β+β_1}}$ is (homogeneously) characterized by a Scott sentence, for all $0<β_1<ω_1$. So, the answer to the above question is posit…
The current paper answers an open question of abs/1007.2426 We say that a countable model M characterizes an infinite cardinal kappa, if the Scott sentence of M has a model in cardinality kappa, but no models in cardinality kappa plus. If M is linearly ordered by <, we will say that the linear ordering (M,<) characterizes kappa. It is known that if kappa is characterizable, then kappa plus is characterizable by a linear ordering. Also, if kappa is characterizable by a dense linear ordering with an increasing sequence of size kappa, then 2^kappa is characterizable. We show that if kappa is homo…
We show that adding a random real number destroys a large fragment of Martin's axiom, namely Martin's axiom for partial orders that have precalibre-$\aleph_1$, thus answering an old question of J. Roitman [9]. We also answer a question of J. Steprans and S. Watson [13] by showing that, by a forcing that preserves cardinals, one can destroy the precalibre-$\aleph_1$ property of a partial ordering while preserving its ccc-ness.
A charge (finitely additive measure) defined on a Boolean algebra of sets taking values in a group $G$ is called a strictly nonzero (SNZ) charge if it takes the identity value in $G$ only for the zero element of the Boolean algebra. A study of such charges was initiated by Rudiger Göbel and K.P.S. Bhaskara Rao in 2002. Our main result is a solution to one of the questions posed in that paper: we show that for every cardinal $\aleph$, the Boolean algebra of clopen sets of $\{0,1\}^\aleph$ has a strictly nonzero integer-valued charge. The key lemma that we prove is that there exists a strictly n…
We modify the game Fuchino, Koppelberg, and Shelah used to characterize the $κ$-Freese-Nation property for a given Boolean algebra $A$, replacing players I and II each with a team of $n$ players with limited information. We show that $A$ is tightly $κ$-filtered exactly when team II has a winning strategy for every finite team size. Case $κ=\aleph_0$ characterizes projective Boolean algebras and, hence, Dugundji spaces. In terms of the open-open game of Daniels, Kunen, and Zhou, this characterization is a team version of very I-favorable. We similarly characterize Cohen algebras in terms of a t…
We study the question which Boolean algebras have the property that for every generating set there is an ultrafilter selecting maximal number of its elements. We call it the ultrafilter selection property. For cardinality aleph-one the property is equivalent to the fact that the space of ultrafilters is not Corson compact. We also consider the pointwise topology on a Boolean algebra, proving a result on the Lindelöf number in the context of the ultrafilter selection property. Finally, we discuss poset Boolean algebras, interval algebras, and semilattices in the context of ultrafilter selection…
An interval algebra is a Boolean algebra which is isomorphic to the algebra of finite unions of half-open intervals, of a linearly ordered set. An interval algebra is hereditary if every subalgebra is an interval algebra. We answer a question of M. Bekkali and S. Todorčević, by showing that it is consistent that every $σ$-centered interval algebra of size $\mathfrak{b}$ is hereditary. We also show that there is, in ZFC, an hereditary interval algebra of cardinality $\aleph_1.$
