Filtration equivalent aleph_1-separable abelian groups of cardinality aleph_1

Order-isomorphic Morass-definable $η_1$-orderings

We prove that in the Cohen extension adding $\aleph_3$ generic reals to a model of $ZFC+CH$ containing a simplified $(ω_1,2)$-morass, gap-2 morass-definable $η_1$-orderings with cardinality $\aleph_3$ are order-isomorphic. Hence it is consistent that the $2^{\aleph_0}=\aleph_3$ and that morass-definable $η_1$-orderings with cardinality of the continuum are order-isomorphic. We prove that there are ultrapowers of $\mathbb{R}$ over $ω$ that are gap-2 morass-definable. The constructions use a simplified gap-2 morass, and commutativity with morass-maps and morass-embeddings, to extend a transfinit…

Strong dichotomy of cardinality

A usual dichotomy is that in many cases, reasonably definable sets, satisfy the CH, i.e. if they are uncountable they have cardinality continuum. A strong dichotomy is when: if the cardinality is infinite it is continuum as in [Sh:273]. We are interested in such phenomena when lambda = aleph_0 is replaced by lambda regular uncountable and also by lambda = beth_omega or more generally by strong limit of cofinality aleph_0 .

Forcing a Basis into $\aleph_1$-Free Groups

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 $Γ(H) \neq [\aleph_1]$, …

Historic iteration with aleph_epsilon-support

One aim of this work is to get a universe in which weak versions of Martin axioms holds for some forcing notions of cardinality aleph_0, aleph_1 and aleph_2 while on aleph_2 club, the ``small'' brother of diamond, holds. As a consequence we get the consistency of ``there is no Gross space''. Another aim is to present a case of ``Historic iteration''.

Uniformization and the diversity of Whitehead groups

The connections between Whitehead groups and uniformization properties were investigated by the third author in [Sh:98]. In particular it was essentially shown there that there is a non-free Whitehead (respectively, aleph_1-coseparable) group of cardinality aleph_1 if and only if there is a ladder system on a stationary subset of omega_1 which satisfies 2-uniformization (respectively, omega-uniformization). These techniques allowed also the proof of various independence and consistency results about Whitehead groups, for example that it is consistent that there is a non-free Whitehead group of…

On coherent systems of projections for aleph_1 separable groups

It is proved consistent with either CH or the negation of CH that there is an aleph_1-separable group of cardinality aleph_1 which does not have a coherent system of projections. It had previously been shown that it is consistent with not CH that every aleph_1-separable group of cardinality aleph_1 does have a coherent system of projections.

The Cardinality of the second uniform indiscernible

When the second uniform indiscernible is $\aleph_{2}$, the Martin-Solovay tree only constructs countably many reals; this resolves a number of open questions in descriptive set theory.

The Ehrenfeucht-Fraisse-game of length omega_1

Let (A) and (B) be two first order structures of the same vocabulary. We shall consider the Ehrenfeucht-Fra{i}sse-game of length omega_1 of A and B which we denote by G_{omega_1}(A,B). This game is like the ordinary Ehrenfeucht-Fraisse-game of L_{omega omega} except that there are omega_1 moves. It is clear that G_{omega_1}(A,B) is determined if A and B are of cardinality <= aleph_1. We prove the following results: Theorem A: If V=L, then there are models A and B of cardinality aleph_2 such that the game G_{omega_1}(A,B) is non-determined. Theorem B: If it is consistent that there is a measura…

低于 $\aleph_ω$ 的共尾类型

Cofinal types below $\aleph_ω$

本文证明基数不超过 $\aleph_n$ 的非 Tukey 等价有向集数量至少为 $(n+2)$-Catalan 数，其 Tukey 类包含 Dyck 路径偏序集的同构副本，并刻画了相继元素间的关系。

Endomorphism rings of modules whose cardinality is cofinal to omega

The main result is Theorem: Let A be an R-algebra, mu, lambda be cardinals such that |A|<=mu=mu^{aleph_0}<lambda<=2^mu. If A is aleph_0-cotorsion-free or A is countably free, respectively, then there exists an aleph_0-cotorsion-free or a separable (reduced, torsion-free) R-module G respectively of cardinality |G|=lambda with End_RG=A oplus Fin G.

Topology, Metric Spaces and the Generalized Continuum Hypothesis

This is a paper that aims to interpret the cardinality of a set in terms of Baire Category, i.e. how many closed nowhere dense sets can be deleted from a set before the set itself becomes negligible. . To do this natural tree-theoretic structures such as the Baire topology are introduced, and the Baire Category Theorem is extended to a statement that a $\aleph$-sequentially complete binary tree representation of a Hausdorff topological space that has a clopen base of cardinality $\aleph$ and no isolated or discrete points is not the union of $<\aleph+1$-many nowhere dense subsets for cardinal …

A model theoretic solution to a problem of László Fuchs

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 $p$-…

Abstract elementary classes near aleph_1

We prove in ZFC, no psi in L_{omega_1,omega}[Q] have unique model of uncountable cardinality, this confirms theBaldwin conjecture. But we analyze this in more general terms. We introduce and investigate a.e.c. and also versions of limit models, and prove some basic properties like representation by PC class, for any a.e.c. For PC_{aleph_0}-representable a.e.c. we investigate the conclusion of having not too many non-isomorphic models in aleph_1 and aleph_2, but have to assume 2^{aleph_0}<2^{aleph_1} and even 2^{aleph_1}<2^{aleph_2}.

GCH implies the existence of many rigid almost free abelian groups

We begin with the existence of groups with trivial duals for cardinals aleph_n (n in omega). Then we derive results about strongly aleph_n-free abelian groups of cardinality aleph_n (n in omega) with prescribed free, countable endomorphism ring. Finally we use combinatorial results of [Sh:108], [Sh:141] to give similar answers for cardinals >aleph_omega. As in Magidor and Shelah [MgSh:204], a paper concerned with the existence of kappa-free, non-free abelian groups of cardinality kappa, the induction argument breaks down at aleph_omega. Recall that aleph_omega is the first singular cardinal an…

A ZFC Dowker space in $\aleph_{ω+1}$: an application of pcf theory to topology

A ZFC Dowker space is constructed which has cardinality $\aleph_{ω+1}$. This provides a bound in ZFC to the first cardinal in which there is a ZFC Dowker space. The space we construct is a closed and cofinal subspace of M.~E.~Rudin's Dowker space from 1971. A theorem from pcf theory used in the proof, but otherwise the proof is elementary.

Spectral subspaces of spectra of Abelian lattice-ordered groups in size aleph one

It is well known that the lattice Idc G of all principal {\ell}-ideals of any Abelian {\ell}-group G is a completely normal distributive 0-lattice, and that not every completely normal distributive 0-lattice is a homomorphic image of some Idc G, via a counterexample of cardinality $\aleph 2. We prove that every completely normal distributive 0-lattice with at most $\aleph 1 elements is a homomorphic image of some Idc G. By Stone duality, this means that every completely normal generalized spectral space, with at most $\aleph 1 compact open sets, is homeomorphic to a spectral subspace of the {\…

Universal in (< lambda)-stable abelian group

A characteristic result is that if 2^{aleph_0}< mu < mu^+< lambda = cf(lambda)< mu^{aleph_0}, then among the separable reduced p-groups of cardinality lambda which are (< lambda)-stable there is no universal one.

New non-free Whitehead groups (corrected version)

We show that it is consistent that there is a strongly aleph_1-free aleph_1-coseparable group of cardinality aleph_1 which is not aleph_1-separable.

On regular reduced products

Assume <aleph_0,aleph_1>-><lambda,lambda^+>. Assume M is a model of a first order theory T of cardinality at most lambda^+ in a vocabulary L(T) of cardinality <= lambda . Let N be a model with the same vocabulary. Let Delta be a set of first order formulas in L(T) and let D be a regular filter on lambda. Then M is Delta-embeddable into the reduced power N^lambda/D, provided that every Delta-existential formula true in M is true also in N. We obtain the following corollary: for M as above and D a regular ultrafilter over lambda, M^lambda/D is lambda^{++}-universal. Our second result is as follo…

Almost disjoint pure subgroups of the Baer-Specker group

We prove in ZFC that the Baer-Specker group Z^omega has 2^{aleph_1} non-free pure subgroups of cardinality aleph_1 which are almost disjoint: there is no non-free subgroup embeddable in any pair.

The Kaplansky test problems for aleph_1-separable groups

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 .

Almost isometric embeddings of metric spaces

We investigate a relations of almost isometric embedding and almost isometry between metric spaces and prove that with respect to these relations: (1) There is a countable universal metric space. (2) There may exist fewer than continuum separable metric spaces on aleph_1 so that every separable metric space is almost isometrically embedded into one of them when the continuum hypothesis fails. (3) There is no collection of fewer than continuum metric spaces of cardinality aleph_2 so that every ultra-metric space of cardinality aleph_2 is almost isometrically embedded into one of them if aleph_2…

弱钻石原则与pcf理论

Weak diamond and pcf theory

本文利用弱钻石原则推导出pcf(𝔞)的基数上界，并证明在温和假设下许多奇异基数ℵ_δ满足2^{ℵ_δ}<ℵ_{(|δ|^{+3})}。

关于最大阶梯

On maximal ladders

本文研究了最大n-阶梯的性质，证明在Add(ω,ω_ω)下，最大n-阶梯的基数为ℵ_{n-1}，并探讨了相关问题的致密性。

On the number of L_{infty,omega_1}-equivalent non-isomorphic models

We prove that if ZF is consistent then ZFC+GCH is consistent with the following statement: There is for every k<omega a model of cardinality aleph_1 which is L_{infty,omega_1}-equivalent to exactly k non-isomorphic models of cardinality aleph_1. In order to get this result we introduce ladder systems and colourings different from the ``standard'' counterparts, and prove the following purely combinatorial result: For each prime number p and positive integer m it is consistent with ZFC+GCH that there is a ``good'' ladder system having exactly p^m pairwise nonequivalent colourings.

具有大正方形的Borel集

Borel sets with large squares

本文修正了早期研究，给出基数相关的充分条件$\oplus_μ$，证明其与Borel集大正方形存在性、无穷逻辑模型存在性的条件在MA假设下等价，并推导相关基数性质，还涉及Borel矩形与模型论问题。

Representing embeddability as set inclusion

A few steps are made towards representation theory of embeddability among uncountable graphs. A monotone class of graphs is defined by forbidding countable subgraphs, related to the graph's end-structure. Using a combinatorial theorem of Shelah it is proved: - The complexity of the class in every regular uncountable $ł>\aleph_1$ is at least $λ^+ + \sup\{μ^{\aleph_0}:μ^+<λ\}$ - For all regular uncountable $λ>\aleph_1$ there are $2^λ$ pairwise non embeddable graphs in the class having strong homogeneity properties. - It is characterized when some invariants of a graph $G\in \Cal G_λ$ have to be …

向量子空间的Mad族与实数的最小非贫集

Mad families of vector subspaces and the smallest nonmeager set of reals

本文证明参数化菱形原理（对应贫理想一致性）蕴含可数向量空间中无限极大几乎不相交块子空间族的最小基数为阿列夫一，进而在Miller模型中该不变量为阿列夫一。

等价理论区间与逻辑的基数

The Cardinalities of Intervals of Equational Theories and Logics

本文利用描述集合论研究等价理论和逻辑类的基数，解决了一些开放问题，证明了某些集合在Borel层次中的位置及其基数。

紧σ-滤过与C(ω*)的BFPP

Tight $σ$-filtrations and the BFPP of $C(ω^*)$

在P(ω)紧σ-滤过的假设下，证明C(ω*)不满足球不动点性质，利用Cc(ω₁)与P(ω)紧σ-滤过性的等价性，还得到相关布尔代数商的性质。

On our paper `Almost Free Splitter', a correction

Let R be a subring of Q and recall from math.LO/9910161 that an R-module G is a splitter if Ext_R(G,G)=0. We correct the statement of Main Theorem 1.5 in math.LO/9910161. Assuming CH any aleph_1$-free splitter of cardinality aleph_1 is free over its nucleus as shown in math.LO/9910161. Generally these modules are very close to being free as explained below. This change follows from math.LO/9910161 and is due to an incomplete proof (noticed thanks to Paul Eklof) in the first section of math.LO/9910161. Assuming the negation of CH, in Shelah [Sh:F417] (work in progress) it will be shown that und…

How to drive our families mad

Given a family $F$ of pairwise almost disjoint sets on a countable set $S$, we study maximal almost disjoint (mad) families $F^+$ extending $F$. We define $a^+(F)$ to be the minimal possible cardinality of $F^+\setminus F$ for such $F^+$, and $a^+(κ)=\sup\{a^+(F): |F| \leq κ\}$. We show that all infinite cardinal less than or equal to the continuum continuum can be represented as $a^+(F)$ for some almost disjoint $F$ and that the inequalities $\aleph_1=a<a^+(\aleph_1)=c$ and $a=a^+(\aleph_1)<c$ are both consistent. We also give a several constructions of mad families with some additional prope…

EF equivalent not isomorphic models

We construct non-isomorphic models M, N, e.g. of cardinality aleph_1 such that in the Ehrenfeucht-Fraisse game of length zeta < omega_1 the isomorphism player wins

Cardinalities of topologies with small base

Let T be the family of open subsets of a topological space (not necessarily Hausdorff or even T_0). We prove that if T has a base of cardinality <= mu, lambda <= mu < 2^lambda, lambda strong limit of cofinality aleph_0, then T has cardinality <= mu or >= 2^lambda. This is our main conclusion. First we prove it under some set theoretic assumption, which is clear when lambda = mu ; then we eliminate the assumption by a theorem on pcf from [Sh 460] motivated originally by this. Next we prove that the simplest examples are the basic ones; they occur in every example (for lambda = aleph_0 this fulf…

On distinguishing quotients of symmetric groups

A study is carried out of the elementary theory of quotients of symmetric groups in a similar spirit to [Sh:24]. Apart from the trivial and alternating subgroups, the normal subgroups of the full symmetric group S(mu) on an infinite cardinal mu are all of the form S_kappa(mu)= the subgroup consisting of elements whose support has cardinality <kappa for some kappa <= mu^+. A many-sorted structure M_{kappa lambda mu} is defined which, it is shown, encapsulates the first order properties of the group S_lambda (mu)/S_kappa (mu). Specifically, these two structures are (uniformly) bi-interpretable, …

Ditor问题的解答

A solution to Ditor's problem

本文证明存在基数为ℵ₂的3-梯子与马洛基数一致，从而解决Ditor提出的长期未决问题，并表明该问题独立于ZFC。

Strong downward Löwenheim-Skolem theorems for stationary logics, II -- reflection down to the continuum

Continuing the previous paper, we study the Strong Downward Löwenheim-Skolem Theorems (SDLSs) of the stationary logic and their variations. It has been shown that the SDLS for the ordinary stationary logic with weak second-order parameters down to $<\aleph_2$ is equivalent to the conjunction of CH and Cox's Diagonal Reflection Principle for internally clubness. We show that the SDLS for the stationary logic without weak second-order parameters down to $<2^{\aleph_0}$ implies that the size of the continuum is $\aleph_2$. In contrast, an internal interpretation of the stationary logic can satisf…

关于局部紧致集的上同调

On cohomology of locally profinite sets

本文在Zermelo-Fraenkel集合论中构造了首个非递降且忠实平坦的交换环之间映射的例子，涉及局部紧致集的上同调结构。

Exercices de style: a homotopy theory for set theory, I

We construct a model category (in the sense of Quillen) for set theory, starting from two arbitrary, but natural, conventions. It is the simplest category satisfying our conventions and modelling the notions of finiteness, countability and infinite equi-cardinality. In a subsequent paper \cite{GaHa1} we give a homotopy theoretic dictionary of set theoretic concepts, most notably Shelah's covering number $\cov(λ, \aleph_1,\aleph_1,2)$, recovered from this model category. We argue that from the homotopy theoretic point of view our construction is essentially automatic following basic existing me…

Free groups and automorphism groups of infinite fields

Let λbe a cardinal with λ=λ^{\aleph_0} and p be either 0 or a prime number. We show that there are fields K_0 and K_1 of cardinality λand characteristic p such that the automorphism group of K_0 is a free group of cardinality 2^λand the automorphism group of K_1 is a free abelian group of cardinality 2^λ. This partially answers a question from [8] and complements results from [15], [16] and [17]. The methods developed in the proof of the above statement also allow us to show that the above cardinal arithmetic assumption is consistently not necessary for the existence of such fields and that th…

The Generalized Continuum Hypothesis revisited

We argue that we solved Hilbert's first problem positively (after reformulating it just to avoid the known consistency results) and give some applications. Let lambda to the revised power of kappa, denoted lambda^{[kappa]}, be the minimal cardinality of a family of subsets of lambda each of cardinality kappa such that any other subset of lambda of cardinality kappa is included in the union of <kappa members of the family. The main theorem says that almost always this revised power is equal to lambda. Our main result is The Revised GCH Theorem: Assume we fix an uncountable strong limit cardinal…

Erdös–Dushnik–Miller定理的无选择证明

A choice-free proof of the Erdős--Dushnik--Miller theorem

本文在不含选择公理的ZF集合论中，给出Erdös–Dushnik–Miller定理的纯组合证明，避免了元数学考量。

介于 Whitehead 群与一致化之间

Between Whitehead groups and uniformization

本文在ZFC中证明了关于可数序数平稳集的梯子系统一致化与强自由阿贝尔群的余可分性等价，解决了Eklof和Mekler专著中的问题B3和B4。

On Non-standard Models of Arithmetic with Uncountable Standard Systems

In 1960s, Dana Scott gave a recursion theoretic characterization of standard systems of countable non-standard models of arithmetic, i.e., collections of sets of standard natural numbers coded in non-standard models. Later, Knight and Nadel proved that Scott's characterization also applies to non-standard models of arithmetic with cardinality $\aleph_1$. But the question, whether the limit on cardinality can be removed from the above characterization, remains a long standing question, known as the Scott Set Problem. This article presents two constructions of non-standard models of arithmetic w…

On cardinalities in quotients of inverse limits of groups

Let lambda be aleph_0 or a strong limit of cofinality aleph_0. Suppose that (G_m,p_{m,n}:m =< n<omega) and (H_m,p^t_{m,n}: m=< n < omega) are projective systems of groups of cardinality less than lambda and suppose that for every n<omega there is a homomorphism h:H_n-->G_n such that all the diagrams commute. If for every mu<lambda there exists (f_i in G_omega:i<mu) such that for distinct i,j we have: f_i f_j^{-1} notin h_omega(H_omega), then there exists (f_i in G_omega:i<2^lambda) such that for distinct i,j we have f_i f_j^{-1} notin h_omega(H_omega).

Positional strategies in long Ehrenfeucht-Fraissé games

We prove that it is relatively consistent with ZF + CH that there exist two models of cardinality \aleph_2 such that the second player has a winning strategy in the Ehrenfeucht-Fraïssé-game of length ω_1 but there is no σ-closed back-and-forth set for the two models. If CH fails, no such pairs of models exist.

Uniquely Transitive Torsion-free Abelian Groups

We will answer a question raised by Emmanuel Dror Farjoun concerning the existence of torsion-free abelian groups G such that for any ordered pair of pure elements there is a unique automorphism mapping the first element onto the second one. We will show the existence of such a group of cardinality lambda for any successor cardinal lambda=mu^+ with mu=mu^{aleph_0}.

On a conjecture regarding nonstandard uniserial modules

We consider the question of which valuation domains (of cardinality aleph_1) have non-standard uniserial modules. We show that a criterion conjectured by Osofsky is independent of ZFC + GCH.

基数阿列夫一下 MV-谱问题的一个解

A solution to the MV-spectrum Problem in size aleph one

本文证明了可数阿贝尔 $\ell$-群主理想格的闭同态提升定理，刻画了至多 $\aleph_1$ 基数的分配 0-格同构条件，从而解决了该基数下的 Mundici MV-谱问题，并将结果推广至森林索引交换图及任意可数全序除环上的向量格。

Bext^2(G,T) can be nontrivial, even assuming GCH

Using the consistency of some large cardinals we produce a model of Set Theory in which the generalized continuum hypothesis holds and for some torsion-free abelian group G of cardinality aleph_{omega +1} and for some torsion group T, Bext^2(G,T) not =0 .

Some Problems in Automata Theory Which Depend on the Models of Set Theory

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…

Co-Hopfian 与有界内自同构刚性的混合阿贝尔群

Co-Hopfian and boundedly endo-rigid mixed abelian groups

该研究在温和基数条件下构造了指定挠子群的有界内自同构刚性阿贝尔群，给出配对完整刻画，应用双重黑箱方法，并解决了大基数下co-Hopfian混合阿贝尔群的存在性问题并给出判定准则。

Every coseparable group may be free

We show that if 2^{aleph_0} Cohen reals are added to the universe, then for every reduced non-free torsion-free abelian group A of cardinality less than the continuum, there is a prime p so that Ext_p(A, Z) not= 0. In particular if it is consistent that there is a supercompact cardinal, then it is consistent (even with weak CH) that every coseparable group is free. The use of some large cardinal hypothesis is needed.

多少喷雾能覆盖空间？

How many sprays cover the space?

本文研究了在超平面中处于一般位置的喷雾数量与实数集基数之间的关系，并扩展了 Schmerl 在二维情况下的研究成果。