P-points, MAD families and Cardinal Invariants

Osvaldo Guzmán González
{"title":"P-points, MAD families and Cardinal Invariants","authors":"Osvaldo Guzmán González","doi":"10.1017/bsl.2021.24","DOIUrl":null,"url":null,"abstract":"Abstract The main topics of this thesis are cardinal invariants, P -points and MAD families. Cardinal invariants of the continuum are cardinal numbers that are bigger than \n$\\aleph _{0}$\n and smaller or equal than \n$\\mathfrak {c}.$\n Of course, they are only interesting when they have some combinatorial or topological definition. An almost disjoint family is a family of infinite subsets of \n$\\omega $\n such that the intersection of any two of its elements is finite. A MAD family is a maximal almost disjoint family. An ultrafilter \n$\\mathcal {U}$\n on \n$\\omega $\n is called a P-point if every countable \n$\\mathcal {B\\subseteq U}$\n there is \n$X\\in $\n \n$\\mathcal {U}$\n such that \n$X\\setminus B$\n is finite for every \n$B\\in \\mathcal {B}.$\n This kind of ultrafilters has been extensively studied, however there is still a large number of open questions about them. In the preliminaries we recall the principal properties of filters, ultrafilters, ideals, MAD families and cardinal invariants of the continuum. We present the construction of Shelah, Mildenberger, Raghavan, and Steprāns of a completely separable MAD family under \n$\\mathfrak {s\\leq a}.$\n None of the results in this chapter are due to the author. The second chapter is dedicated to a principle of Sierpiński. The principle \n$\\left ( \\ast \\right ) $\n of Sierpiński is the following statement: There is a family of functions \n$\\left \\{ \\varphi _{n}:\\omega _{1}\\longrightarrow \\omega _{1}\\mid n\\in \\omega \\right \\} $\n such that for every \n$I\\in \\left [ \\omega _{1}\\right ] ^{\\omega _{1}}$\n there is \n$n\\in \\omega $\n for which \n$\\varphi _{n}\\left [ I\\right ] =\\omega _{1}.$\n This principle was recently studied by Arnie Miller. He showed that this principle is equivalent to the following statement: There is a set \n$X=\\left \\{ f_{\\alpha }\\mid \\alpha <\\omega _{1}\\right \\} \\subseteq \\omega ^{\\omega }$\n such that for every \n$g:\\omega \\longrightarrow \\omega $\n there is \n$\\alpha $\n such that if \n$\\beta>\\alpha $\n then \n$f_{\\beta }\\cap g$\n is infinite (sets with that property are referred to as \n$\\mathcal {IE}$\n -Luzin sets ). Miller showed that the principle of Sierpiński implies that non \n$\\left ( \\mathcal {M}\\right ) =\\omega _{1}.$\n He asked if the converse was true, i.e., does non \n$\\left ( \\mathcal {M}\\right ) =\\omega _{1}$\n imply the principle \n$\\left ( \\ast \\right ) $\n of Sierpiński? We answer his question affirmatively. In other words, we show that non \n$\\left ( \\mathcal {M}\\right ) =\\omega _{1}$\n is enough to construct an \n$\\mathcal {IE}$\n -Luzin set. It is not hard to see that the \n$\\mathcal {IE}$\n -Luzin set we constructed is meager. This is no coincidence, because with the aid of an inaccessible cardinal, we construct a model where non \n$\\left ( \\mathcal {M}\\right ) =\\omega _{1}$\n and every \n$\\mathcal {IE}$\n -Luzin set is meager. The third chapter is dedicated to a conjecture of Hrušák. Michael Hrušák conjectured the following: Every Borel cardinal invariant is either at most non \n$\\left ( \\mathcal {M}\\right ) $\n or at least cov \n$\\left ( \\mathcal {M}\\right ) $\n (it is known that the definability is an important requirement, otherwise \n$\\mathfrak {a}$\n would be a counterexample). Although the veracity of this conjecture is still an open problem, we were able to obtain some partial results: The conjecture is false for “Borel invariants of \n$\\omega _{1}^{\\omega }$\n ” nevertheless, it is true for a large class of definable invariants. This is part of a joint work with Michael Hrušák and Jindřich Zapletal. In the fourth chapter we present a survey on destructibility of ideals and MAD families. We prove several classic theorems, but we also prove some new results. For example, we show that every almost disjoint family of size less than \n$\\mathfrak {c}$\n can be extended to a Cohen indestructible MAD family is equivalent to \n$\\mathfrak {b=c}$\n (this is part of a joint work with Michael Hrušák, Ariet Ramos, and Carlos Martínez). A MAD family \n$\\mathcal {A}$\n is Shelah–Steprāns if for every \n$X\\subseteq \\left [ \\omega \\right ] ^{<\\omega }\\setminus \\left \\{ \\emptyset \\right \\} $\n either there is \n$A\\in \\mathcal {I}\\left ( \\mathcal {A}\\right ) $\n such that \n$s\\cap A\\neq \\emptyset $\n for every \n$s\\in X$\n or there is \n$B\\in \\mathcal {I}\\left ( \\mathcal {A}\\right ) $\n that contains infinitely many elements of X (where \n$\\mathcal {I}\\left ( \\mathcal {A}\\right ) $\n denotes the ideal generated by \n$\\mathcal {A}$\n ). This concept was introduced by Raghavan which is connected to the notion of “strongly separable” introduced by Shelah and Steprāns. We prove that Shelah–Steprāns MAD families have very strong indestructibility properties: Shelah–Steprāns MAD families are indestructible for “many” definable forcings that does not add dominating reals (this statement will be formalized in the fourth chapter). According to the author’s best knowledge, this is the strongest notion (in terms of indestructibility) that has been considered in the literature so far. In spite of their strong indestructibility, Shelah–Steprāns MAD families can be destroyed by a ccc forcing that does not add unsplit or dominating reals. We also consider some strong combinatorial properties of MAD families and show the relationships between them (This is part of a joint work with Michael Hrušák, Dilip Raghavan, and Joerg Brendle). The fifth chapter is one of the most important chapters in the thesis. A MAD family \n$\\mathcal {A}$\n is called \n$+$\n -Ramsey if every tree that branches into \n$\\mathcal {I}\\left ( \\mathcal {A}\\right ) $\n -positive sets has an \n$\\mathcal {I}\\left ( \\mathcal {A}\\right ) $\n -positive branch. Michael Hrušák’s first published question is the following: Is there a \n$+$\n -Ramsey MAD family? It was previously known that such families can consistently exist. However, there was no construction of such families using only the axioms of ZFC. We solve this problem by constructing such a family without any extra assumptions. In the fourth and fifth chapters, we introduce several notions of MAD families, in the sixth chapter we prove several implications and non implications between them. We construct (under \n$\\mathsf {CH}$\n ) several MAD families with different properties. In the seventh chapter we build models without P -points. We show that there are no P -points after adding Silver reals either iteratively or by the side by side product. These results have some important consequences: The first one is that is its possible to get rid of P -points using only definable forcings. This answers a question of Michael Hrušák. We can also use our results to build models with no P -points and with arbitrarily large continuum, which was also an open question. These results were obtained with David Chodounský. Abstract prepared by Osvaldo Guzmán González E-mail : oguzman@matmor.unam.mx URL : https://arxiv.org/abs/1810.09680","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"The Bulletin of Symbolic Logic","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1017/bsl.2021.24","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0

Abstract

Abstract The main topics of this thesis are cardinal invariants, P -points and MAD families. Cardinal invariants of the continuum are cardinal numbers that are bigger than $\aleph _{0}$ and smaller or equal than $\mathfrak {c}.$ Of course, they are only interesting when they have some combinatorial or topological definition. An almost disjoint family is a family of infinite subsets of $\omega $ such that the intersection of any two of its elements is finite. A MAD family is a maximal almost disjoint family. An ultrafilter $\mathcal {U}$ on $\omega $ is called a P-point if every countable $\mathcal {B\subseteq U}$ there is $X\in $ $\mathcal {U}$ such that $X\setminus B$ is finite for every $B\in \mathcal {B}.$ This kind of ultrafilters has been extensively studied, however there is still a large number of open questions about them. In the preliminaries we recall the principal properties of filters, ultrafilters, ideals, MAD families and cardinal invariants of the continuum. We present the construction of Shelah, Mildenberger, Raghavan, and Steprāns of a completely separable MAD family under $\mathfrak {s\leq a}.$ None of the results in this chapter are due to the author. The second chapter is dedicated to a principle of Sierpiński. The principle $\left ( \ast \right ) $ of Sierpiński is the following statement: There is a family of functions $\left \{ \varphi _{n}:\omega _{1}\longrightarrow \omega _{1}\mid n\in \omega \right \} $ such that for every $I\in \left [ \omega _{1}\right ] ^{\omega _{1}}$ there is $n\in \omega $ for which $\varphi _{n}\left [ I\right ] =\omega _{1}.$ This principle was recently studied by Arnie Miller. He showed that this principle is equivalent to the following statement: There is a set $X=\left \{ f_{\alpha }\mid \alpha <\omega _{1}\right \} \subseteq \omega ^{\omega }$ such that for every $g:\omega \longrightarrow \omega $ there is $\alpha $ such that if $\beta>\alpha $ then $f_{\beta }\cap g$ is infinite (sets with that property are referred to as $\mathcal {IE}$ -Luzin sets ). Miller showed that the principle of Sierpiński implies that non $\left ( \mathcal {M}\right ) =\omega _{1}.$ He asked if the converse was true, i.e., does non $\left ( \mathcal {M}\right ) =\omega _{1}$ imply the principle $\left ( \ast \right ) $ of Sierpiński? We answer his question affirmatively. In other words, we show that non $\left ( \mathcal {M}\right ) =\omega _{1}$ is enough to construct an $\mathcal {IE}$ -Luzin set. It is not hard to see that the $\mathcal {IE}$ -Luzin set we constructed is meager. This is no coincidence, because with the aid of an inaccessible cardinal, we construct a model where non $\left ( \mathcal {M}\right ) =\omega _{1}$ and every $\mathcal {IE}$ -Luzin set is meager. The third chapter is dedicated to a conjecture of Hrušák. Michael Hrušák conjectured the following: Every Borel cardinal invariant is either at most non $\left ( \mathcal {M}\right ) $ or at least cov $\left ( \mathcal {M}\right ) $ (it is known that the definability is an important requirement, otherwise $\mathfrak {a}$ would be a counterexample). Although the veracity of this conjecture is still an open problem, we were able to obtain some partial results: The conjecture is false for “Borel invariants of $\omega _{1}^{\omega }$ ” nevertheless, it is true for a large class of definable invariants. This is part of a joint work with Michael Hrušák and Jindřich Zapletal. In the fourth chapter we present a survey on destructibility of ideals and MAD families. We prove several classic theorems, but we also prove some new results. For example, we show that every almost disjoint family of size less than $\mathfrak {c}$ can be extended to a Cohen indestructible MAD family is equivalent to $\mathfrak {b=c}$ (this is part of a joint work with Michael Hrušák, Ariet Ramos, and Carlos Martínez). A MAD family $\mathcal {A}$ is Shelah–Steprāns if for every $X\subseteq \left [ \omega \right ] ^{<\omega }\setminus \left \{ \emptyset \right \} $ either there is $A\in \mathcal {I}\left ( \mathcal {A}\right ) $ such that $s\cap A\neq \emptyset $ for every $s\in X$ or there is $B\in \mathcal {I}\left ( \mathcal {A}\right ) $ that contains infinitely many elements of X (where $\mathcal {I}\left ( \mathcal {A}\right ) $ denotes the ideal generated by $\mathcal {A}$ ). This concept was introduced by Raghavan which is connected to the notion of “strongly separable” introduced by Shelah and Steprāns. We prove that Shelah–Steprāns MAD families have very strong indestructibility properties: Shelah–Steprāns MAD families are indestructible for “many” definable forcings that does not add dominating reals (this statement will be formalized in the fourth chapter). According to the author’s best knowledge, this is the strongest notion (in terms of indestructibility) that has been considered in the literature so far. In spite of their strong indestructibility, Shelah–Steprāns MAD families can be destroyed by a ccc forcing that does not add unsplit or dominating reals. We also consider some strong combinatorial properties of MAD families and show the relationships between them (This is part of a joint work with Michael Hrušák, Dilip Raghavan, and Joerg Brendle). The fifth chapter is one of the most important chapters in the thesis. A MAD family $\mathcal {A}$ is called $+$ -Ramsey if every tree that branches into $\mathcal {I}\left ( \mathcal {A}\right ) $ -positive sets has an $\mathcal {I}\left ( \mathcal {A}\right ) $ -positive branch. Michael Hrušák’s first published question is the following: Is there a $+$ -Ramsey MAD family? It was previously known that such families can consistently exist. However, there was no construction of such families using only the axioms of ZFC. We solve this problem by constructing such a family without any extra assumptions. In the fourth and fifth chapters, we introduce several notions of MAD families, in the sixth chapter we prove several implications and non implications between them. We construct (under $\mathsf {CH}$ ) several MAD families with different properties. In the seventh chapter we build models without P -points. We show that there are no P -points after adding Silver reals either iteratively or by the side by side product. These results have some important consequences: The first one is that is its possible to get rid of P -points using only definable forcings. This answers a question of Michael Hrušák. We can also use our results to build models with no P -points and with arbitrarily large continuum, which was also an open question. These results were obtained with David Chodounský. Abstract prepared by Osvaldo Guzmán González E-mail : oguzman@matmor.unam.mx URL : https://arxiv.org/abs/1810.09680
p点、MAD族与基数不变量
摘要本文主要讨论基数不变量、P点和MAD族。连续统的基数不变量是大于 $\aleph _{0}$ 小于或等于 $\mathfrak {c}.$ 当然,只有当它们有一些组合或拓扑定义时,它们才有趣。几乎不相交族是由无穷个子集组成的族 $\omega $ 使得任意两个元素的交点是有限的。MAD家族是一个极大的几乎不相交的家族。超滤机 $\mathcal {U}$ on $\omega $ 称为p点,如果每个可数 $\mathcal {B\subseteq U}$ 有 $X\in $ $\mathcal {U}$ 这样 $X\setminus B$ 是有限的 $B\in \mathcal {B}.$ 这种超滤材料已经得到了广泛的研究,但仍有大量的问题有待解决。在序言中,我们回顾了滤光器、超滤光器、理想、MAD族和连续体的基本不变量的主要性质。我们提出了一个完全可分离的MAD家族的Shelah, Mildenberger, Raghavan和Steprāns的构建 $\mathfrak {s\leq a}.$ 本章的结果都不是作者的功劳。第二章论述了Sierpiński的原理。原理 $\left ( \ast \right ) $ Sierpiński的表达式如下:有一个函数族 $\left \{ \varphi _{n}:\omega _{1}\longrightarrow \omega _{1}\mid n\in \omega \right \} $ 这样对于每一个 $I\in \left [ \omega _{1}\right ] ^{\omega _{1}}$ 有 $n\in \omega $ 为了什么? $\varphi _{n}\left [ I\right ] =\omega _{1}.$ 阿尼·米勒最近研究了这个原理。他证明了这个原理等价于下面的陈述:有一个集合 $X=\left \{ f_{\alpha }\mid \alpha \alpha $ 然后 $f_{\beta }\cap g$ 具有该属性的无限集是否被称为 $\mathcal {IE}$ -Luzin sets)。米勒表明Sierpiński原理暗示了非 $\left ( \mathcal {M}\right ) =\omega _{1}.$ 他问反之是否为真,即否 $\left ( \mathcal {M}\right ) =\omega _{1}$ 隐含原则 $\left ( \ast \right ) $ Sierpiński?我们肯定地回答他的问题。换句话说,我们证明了非 $\left ( \mathcal {M}\right ) =\omega _{1}$ 足够构造一个吗 $\mathcal {IE}$ -Luzin set。不难看出, $\mathcal {IE}$ -我们建造的luzin布景很简陋。这不是巧合,因为借助不可接近的基数,我们构建了一个模型 $\left ( \mathcal {M}\right ) =\omega _{1}$ 每一个 $\mathcal {IE}$ -Luzin设置是贫乏的。第三章是关于Hrušák的一个猜想。Michael Hrušák推测如下:每个Borel基本不变量要么最多是非 $\left ( \mathcal {M}\right ) $ 或者至少是cov $\left ( \mathcal {M}\right ) $ (众所周知,可定义性是一个重要的要求,否则 $\mathfrak {a}$ 这是一个反例)。虽然这个猜想的准确性仍然是一个开放的问题,但我们能够得到一些部分结果:对于的Borel不变量,这个猜想是假的 $\omega _{1}^{\omega }$ 然而,对于一大类可定义不变量,它是成立的。这是Michael Hrušák和Jindřich Zapletal共同工作的一部分。第四章对理想与MAD家庭的可破坏性进行了考察。我们证明了几个经典定理,但我们也证明了一些新的结果。例如,我们展示了每一个大小小于 $\mathfrak {c}$ 是否可以延伸到一个科恩坚不可摧的MAD家族 $\mathfrak {b=c}$ (这是与Michael Hrušák、Ariet Ramos和Carlos Martínez共同工作的一部分)。疯狂的家庭 $\mathcal {A}$ 是Shelah-Steprāns if for every $X\subseteq \left [ \omega \right ] ^{<\omega }\setminus \left \{ \emptyset \right \} $ 要么有 $A\in \mathcal {I}\left ( \mathcal {A}\right ) $ 这样 $s\cap A\neq \emptyset $ 对于每一个 $s\in X$ 或者有 $B\in \mathcal {I}\left ( \mathcal {A}\right ) $ 它包含无穷多个X的元素(其中 $\mathcal {I}\left ( \mathcal {A}\right ) $ 表示由生成的理想 $\mathcal {A}$ ). 这个概念是由Raghavan提出的,它与Shelah和Steprāns提出的“强可分离”概念有关。我们证明Shelah-Steprāns MAD族具有很强的不可摧毁性:Shelah-Steprāns MAD族对于“许多”不加支配实数的可定义力是不可摧毁的(这一陈述将在第四章形式化)。据作者所知,这是迄今为止文献中考虑过的最强有力的概念(就不可摧毁性而言)。尽管它们具有强大的不可摧毁性,Shelah-Steprāns MAD家庭可以被一种不增加不分裂或主导现实的ccc强迫所摧毁。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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