Reflexive abelian groups and measurable cardinals and saturated MAD families

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 $p$-…

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…

The least weakly compact cardinal can be unfoldable, weakly measurable and nearly $θ$-supercompact

We prove from suitable large cardinal hypotheses that the least weakly compact cardinal can be unfoldable, weakly measurable and even nearly $θ$-supercompact, for any desired $θ$. In addition, we prove several global results showing how the entire class of weakly compact cardinals, a proper class, can be made to coincide with the class of unfoldable cardinals, with the class of weakly measurable cardinals or with the class of nearly $θ_κ$-supercompact cardinals $κ$, for nearly any desired function $κ\mapstoθ_κ$. These results answer several questions that had been open in the literature and ex…

Madness and regularity properties

Starting from an inaccessible cardinal, we construct a model of $ZF+DC$ where there exists a mad family and all sets of reals are $\mathbb Q$-measurable for $ω^ω$-bounding sufficiently absolute forcing notions $\mathbb Q$.

On the non-existence of $κ$-mad families

Starting from a model with a Laver-indestructible supercompact cardinal $κ$, we construct a model of $ZF+DC_κ$ where there are no $κ$-mad families.

Definable MAD families and forcing axioms

We show that under the Bounded Proper Forcing Axiom and an anti-large cardinal assumption, there is a $\mathbfΠ^1_2$ MAD family.

MAD Families of Projections on l^2 and Real-Valued Functions on Omega

Two sets are said to be almost disjoint if their intersection is finite. Almost disjoint subsets of [omega]^omega and omega^omega have been studied for quite some time. In particular, the cardinal invariants a and a_e, defined to be the minimum cardinality of a maximal infinite almost disjoint family of [omega]^omega and omega^omega respectively, are known to be consistently less than continuum. Here we examine analogs for functions in R^omega and projections on l^2, showing that they too can be consistently less than continuum.

可计算函数的最大最终不同族

Maximal Eventually Different Families of Computable Functions

研究可计算函数的最大最终不同（MED）族的质量问题，通过定义MED族质量问题并与Lempp等人定义的比较，借助Greenberg等人的框架，展示Schrittesser构建的有效封闭MED族，得出相关结果。

Mad Spectra

The mad spectrum is the set of all cardinalities of infinite maximal almost disjoint families on omega. We treat the problem to characterize those sets A which, in some forcing extension of the universe, can be the mad spectrum. We solve this problem to some extent. What remains open is the possible values of min(A) and max(A).

Are a and d your cup of tea?

We show that, consistently, every MAD family has cardinality strictly bigger than the dominating number, that is a > d, thus solving one of the oldest problems on cardinal invariants of the continuum. The method is a contribution to the theory of iterated forcing for making the continuum large.

Higher random indestructibility of MAD families

We give a combinatorial characterization of when a maximal almost disjoint family of a weakly compact cardinal $κ$ is indestructible by the higher random forcing $\mathbb Q_κ$. We then use this characterisation to show that $\mathrm{add}(\mathbf{null}_κ) = \mathfrak b_κ= \mathfrak c_κ$ implies the existence $\mathbb Q_κ$-indestructible family. The results and proofs presented here are parallel to those for classical random forcing.

On the definability of mad families of vector spaces

We consider the definability of mad families in vector spaces of the form $\underset{n<ω}{\bigoplus} F$ where $F$ is a field of cardinality $\leq \aleph_0$. We show that there is no analytic mad family of subspaces when $F=\mathbb{F}_2$, partially answering a question of Smythe. Our proof relies on a variant of Mathias forcing restricted to a certain idempotent ultrafilter whose existence follows from Glazer's proof of Hindman's theorem.

好的投射见证

Good projective witnesses

本文提出一种新的力迫概念，证明极大余有限群的最小基数可严格介于ℵ₁与𝔠之间，且存在最优的Π¹₂-可定义MCG，并在MAD族中建立了类似结果。

是否存在一个巴拿赫空间的“小基数模型”，其对偶空间属于Stegall类但不具有弱*可碎片化性质，还是大基数是必需的？

Are there "small cardinal models" of a Banach space, whose dual space is in Stegall`s class, but it is not $weak^{*}$-fragmentable, or large cardinals are a must?

本文探讨巴拿赫空间对偶空间中弱*可碎片化类与Stegall类是否可能在小基数模型下相等，并指出大基数存在性可导致二者严格真包含。

大基数之上低层次可定义性

Low level definability above large cardinals

研究大基数与可定义性之间的关系，证明了某些基数下的良序和族的存在性，以及内模型的构造。

向量空间的完整mad族与两种局部Ramsey理论

Full mad families of vector spaces and two local Ramsey theories

本文在ZFC下构造了块子空间的完整mad族，解决了Smythe的问题，并讨论了抽象Mathiasforcing在Solovay模型中的应用，证明了在Mahlo基数坍缩模型中不存在此类族。

Cofinitary Groups and Other Almost Disjoint Families of Reals

We study two different types of (maximal) almost disjoint families: very mad families and (maximal) cofinitary groups. For the very mad families we prove the basic existence results. We prove that MA implies there exist many pairwise orthogonal families, and that CH implies that for any very mad family there is one orthogonal to it. Finally we prove that the axiom of constructibility implies that there exists a coanalytic very mad family. Cofinitary groups have a natural action on the natural numbers. We prove that a maximal cofinitary group cannot have infinitely many orbits under this action…

A variant proof of Con(b<a)

We present a variation of the proof in the first author's "Mob families and mad families" of Con(b<a), which in particular removes some of the obstacles to generalising the argument to uncountable cardinals.

Coding into K by reasonable forcing

Assuming that there is no inner model with a strong cardinal, the following is shown: any subset of ω_1 can be made Δ^1_3 (in the codes) by a reasonable set-forcing; there is a reasonable set-generic extension with a Δ^1_3 well-ordering of its reals; 2-step Σ^1_4 absoluteness fails w.r.t. set-sized reasonable forcings.

陡的不可数群

Steep uncountable groups

本研究构造了首个不依赖连续统假设的基数为$\beth_1$的不可数单群，它兼具Artinian、Burnside律、Jónsson性和强有界性，还可实现子群皆单且非平凡子群均malnormal的性质。

Maximal discrete sets

We survey results regarding the definability and size of maximal discrete sets in analytic hypergraphs. Our main examples include maximal almost disjoint (or mad) families, $\mathcal I$-mad families, maximal eventually different families, and maximal cofinitary groups. We discuss the non-increasing sequence of cardinal characteristics $\mathfrak a_ξ$, for $ξ<ω_1$ as well as the notions of spectra of characteristics and optimal projective witnesses. We give an account of Zhang's forcing to add generic cofinitary permutations, and of a version of this forcing with built-in coding.

Gowers 无限块序列的 mad 族

Mad families of Gowers' infinite block sequences

本文基于 Gowers 的 $\mathbf{FIN}_k$ 定理所保证的小子集理想，研究 $\mathbf{FIN}_k$ 上相对于该理想的无限 mad 族，并证明其最小基数 $\mathfrak{a}_{\mathbf{FIN}_k}$ 不可数。

函数投影与复杂度归约

Functional Projection and Complexity Reduction

针对现有复杂度归约证明依赖特定均匀化、难以提升至高阶射影层级的问题，引入弱于均匀化的函数投影原则，证明其可满足多数复杂度归约证明，并在可测基数的典范内模型中成立，进而得到相关组合族的复杂度归约结果。