The Combinatorics and Absoluteness of Definable Sets of Real Numbers

Zach Norwood
{"title":"The Combinatorics and Absoluteness of Definable Sets of Real Numbers","authors":"Zach Norwood","doi":"10.1017/bsl.2021.55","DOIUrl":null,"url":null,"abstract":"Abstract This thesis divides naturally into two parts, each concerned with the extent to which the theory of \n$L(\\mathbf {R})$\n can be changed by forcing. The first part focuses primarily on applying generic-absoluteness principles to how that definable sets of reals enjoy regularity properties. The work in Part I is joint with Itay Neeman and is adapted from our paper Happy and mad families in \n$L(\\mathbf {R})$\n , JSL, 2018. The project was motivated by questions about mad families, maximal families of infinite subsets of \n$\\omega $\n of which any two have only finitely many members in common. We begin, in the spirit of Mathias, by establishing (Theorem 2.8) a strong Ramsey property for sets of reals in the Solovay model, giving a new proof of Törnquist’s theorem that there are no infinite mad families in the Solovay model. In Chapter 3 we stray from the main line of inquiry to briefly study a game-theoretic characterization of filters with the Baire Property. Neeman and Zapletal showed, assuming roughly the existence of a proper class of Woodin cardinals, that the boldface theory of \n$L(\\mathbf {R})$\n cannot be changed by proper forcing. They call their result the Embedding Theorem, because they conclude that in fact there is an elementary embedding from the \n$L(\\mathbf {R})$\n of the ground model to that of the proper forcing extension. With a view toward analyzing mad families under \n$\\mathsf {AD}^+$\n and in \n$L(\\mathbf {R})$\n under large-cardinal hypotheses, in Chapter 4 we establish triangular versions of the Embedding Theorem. These are enough for us to use Mathias’s methods to show (Theorem 4.5) that there are no infinite mad families in \n$L(\\mathbf {R})$\n under large cardinals and (Theorem 4.9) that \n$\\mathsf {AD}^+$\n implies that there are no infinite mad families. These are again corollaries of theorems about strong Ramsey properties under large-cardinal assumptions and \n$\\mathsf {AD}^+$\n , respectively. Our first theorem improves the large-cardinal assumption under which Todorcevic established the nonexistence of infinite mad families in \n$L(\\mathbf {R})$\n . Part I concludes with Chapter 5, a short list of open questions. In the second part of the thesis, we undertake a finer analysis of the Embedding Theorem and its consistency strength. Schindler found that the the Embedding Theorem is consistent relative to much weaker assumptions than the existence of Woodin cardinals. He defined remarkable cardinals, which can exist even in L, and showed that the Embedding Theorem is equiconsistent with the existence of a remarkable cardinal. His theorem resembles a theorem of Harrington–Shelah and Kunen from the 1980s: the absoluteness of the theory of \n$L(\\mathbf {R})$\n to ccc forcing extensions is equiconsistent with a weakly compact cardinal. Joint with Itay Neeman, we improve Schindler’s theorem by showing that absoluteness for \n$\\sigma $\n -closed \n$\\ast $\n ccc posets—instead of the larger class of proper posets—implies the remarkability of \n$\\aleph _1^V$\n in L. This requires a fundamental change in the proof, since Schindler’s lower-bound argument uses Jensen’s reshaping forcing, which, though proper, need not be \n$\\sigma $\n -closed \n$\\ast $\n ccc in that context. Our proof bears more resemblance to that of Harrington–Shelah than to Schindler’s. The proof of Theorem 6.2 splits naturally into two arguments. In Chapter 7 we extend the Harrington–Shelah method of coding reals into a specializing function to allow for trees with uncountable levels that may not belong to L. This culminates in Theorem 7.4, which asserts that if there are \n$X\\subseteq \\omega _1$\n and a tree \n$T\\subseteq \\omega _1$\n of height \n$\\omega _1$\n such that X is codable along T (see Definition 7.3), then \n$L(\\mathbf {R})$\n -absoluteness for ccc posets must fail. We complete the argument in Chapter 8, where we show that if in any \n$\\sigma $\n -closed extension of V there is no \n$X\\subseteq \\omega _1$\n codable along a tree T, then \n$\\aleph _1^V$\n must be remarkable in L. In Chapter 9 we review Schindler’s proof of generic absoluteness from a remarkable cardinal to show that the argument gives a level-by-level upper bound: a strongly \n$\\lambda ^+$\n -remarkable cardinal is enough to get \n$L(\\mathbf {R})$\n -absoluteness for \n$\\lambda $\n -linked proper posets. Chapter 10 is devoted to partially reversing the level-by-level upper bound of Chapter 9. Adapting the methods of Neeman, Hierarchies of forcing axioms II, we are able to show that \n$L(\\mathbf {R})$\n -absoluteness for \n$\\left |\\mathbf {R}\\right |\\cdot \\left |\\lambda \\right |$\n -linked posets implies that the interval \n$[\\aleph _1^V,\\lambda ]$\n is \n$\\Sigma ^2_1$\n -remarkable in L. Abstract prepared by Zach Norwood. 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引用次数: 0

Abstract

Abstract This thesis divides naturally into two parts, each concerned with the extent to which the theory of $L(\mathbf {R})$ can be changed by forcing. The first part focuses primarily on applying generic-absoluteness principles to how that definable sets of reals enjoy regularity properties. The work in Part I is joint with Itay Neeman and is adapted from our paper Happy and mad families in $L(\mathbf {R})$ , JSL, 2018. The project was motivated by questions about mad families, maximal families of infinite subsets of $\omega $ of which any two have only finitely many members in common. We begin, in the spirit of Mathias, by establishing (Theorem 2.8) a strong Ramsey property for sets of reals in the Solovay model, giving a new proof of Törnquist’s theorem that there are no infinite mad families in the Solovay model. In Chapter 3 we stray from the main line of inquiry to briefly study a game-theoretic characterization of filters with the Baire Property. Neeman and Zapletal showed, assuming roughly the existence of a proper class of Woodin cardinals, that the boldface theory of $L(\mathbf {R})$ cannot be changed by proper forcing. They call their result the Embedding Theorem, because they conclude that in fact there is an elementary embedding from the $L(\mathbf {R})$ of the ground model to that of the proper forcing extension. With a view toward analyzing mad families under $\mathsf {AD}^+$ and in $L(\mathbf {R})$ under large-cardinal hypotheses, in Chapter 4 we establish triangular versions of the Embedding Theorem. These are enough for us to use Mathias’s methods to show (Theorem 4.5) that there are no infinite mad families in $L(\mathbf {R})$ under large cardinals and (Theorem 4.9) that $\mathsf {AD}^+$ implies that there are no infinite mad families. These are again corollaries of theorems about strong Ramsey properties under large-cardinal assumptions and $\mathsf {AD}^+$ , respectively. Our first theorem improves the large-cardinal assumption under which Todorcevic established the nonexistence of infinite mad families in $L(\mathbf {R})$ . Part I concludes with Chapter 5, a short list of open questions. In the second part of the thesis, we undertake a finer analysis of the Embedding Theorem and its consistency strength. Schindler found that the the Embedding Theorem is consistent relative to much weaker assumptions than the existence of Woodin cardinals. He defined remarkable cardinals, which can exist even in L, and showed that the Embedding Theorem is equiconsistent with the existence of a remarkable cardinal. His theorem resembles a theorem of Harrington–Shelah and Kunen from the 1980s: the absoluteness of the theory of $L(\mathbf {R})$ to ccc forcing extensions is equiconsistent with a weakly compact cardinal. Joint with Itay Neeman, we improve Schindler’s theorem by showing that absoluteness for $\sigma $ -closed $\ast $ ccc posets—instead of the larger class of proper posets—implies the remarkability of $\aleph _1^V$ in L. This requires a fundamental change in the proof, since Schindler’s lower-bound argument uses Jensen’s reshaping forcing, which, though proper, need not be $\sigma $ -closed $\ast $ ccc in that context. Our proof bears more resemblance to that of Harrington–Shelah than to Schindler’s. The proof of Theorem 6.2 splits naturally into two arguments. In Chapter 7 we extend the Harrington–Shelah method of coding reals into a specializing function to allow for trees with uncountable levels that may not belong to L. This culminates in Theorem 7.4, which asserts that if there are $X\subseteq \omega _1$ and a tree $T\subseteq \omega _1$ of height $\omega _1$ such that X is codable along T (see Definition 7.3), then $L(\mathbf {R})$ -absoluteness for ccc posets must fail. We complete the argument in Chapter 8, where we show that if in any $\sigma $ -closed extension of V there is no $X\subseteq \omega _1$ codable along a tree T, then $\aleph _1^V$ must be remarkable in L. In Chapter 9 we review Schindler’s proof of generic absoluteness from a remarkable cardinal to show that the argument gives a level-by-level upper bound: a strongly $\lambda ^+$ -remarkable cardinal is enough to get $L(\mathbf {R})$ -absoluteness for $\lambda $ -linked proper posets. Chapter 10 is devoted to partially reversing the level-by-level upper bound of Chapter 9. Adapting the methods of Neeman, Hierarchies of forcing axioms II, we are able to show that $L(\mathbf {R})$ -absoluteness for $\left |\mathbf {R}\right |\cdot \left |\lambda \right |$ -linked posets implies that the interval $[\aleph _1^V,\lambda ]$ is $\Sigma ^2_1$ -remarkable in L. Abstract prepared by Zach Norwood. E-mail: zachnorwood@gmail.com
实数可定义集的组合性与绝对性
本文自然分为两部分,每一部分都涉及$L(\mathbf {R})$理论在多大程度上可以被强迫改变。第一部分主要关注于将泛型绝对原理应用于可定义实数集如何享有正则性。第一部分的工作是与Itay Neeman合作,改编自我们在$L(\mathbf {R})$, JSL, 2018年的论文《快乐与疯狂的家庭》。这个项目的动机是关于疯狂家庭的问题,疯狂家庭是$\omega $的无限子集的最大家庭,其中任何两个都只有有限的共同成员。我们以Mathias的精神开始,通过建立(定理2.8)Solovay模型中实数集的强Ramsey性质,给出Törnquist定理的新证明,即Solovay模型中不存在无限的疯狂族。在第三章中,我们偏离了研究的主线,简单地研究了具有贝尔性质的滤波器的博弈论表征。Neeman和Zapletal在大致假设存在一种合适的伍丁基数的情况下,证明了$L(\mathbf {R})$的黑体字理论不能通过适当的强迫来改变。他们把他们的结果称为嵌入定理,因为他们得出结论,事实上,从地面模型的$L(\mathbf {R})$到固有强迫扩展的存在一个初等嵌入。为了分析在$\mathsf {AD}^+$和$L(\mathbf {R})$大基数假设下的疯狂家庭,在第四章中我们建立了嵌入定理的三角版本。这些足以让我们使用Mathias的方法来证明(定理4.5)在$L(\mathbf {R})$中在大基数下不存在无限的疯狂族,并且(定理4.9)$\mathsf {AD}^+$暗示不存在无限的疯狂族。这些分别是在大基数假设和$\mathsf {AD}^+$下关于强拉姆齐性质的定理的推论。我们的第一个定理改进了托多切维奇在$L(\mathbf {R})$中建立无限疯狂家族不存在的大基数假设。第一部分以第5章结束,这是一个简短的开放性问题列表。在论文的第二部分,我们对嵌入定理及其一致性强度进行了更细致的分析。Schindler发现,相对于Woodin基数的存在,嵌入定理在更弱的假设下是一致的。他定义了即使在L中也可以存在的显著基数,并证明了嵌入定理与显著基数的存在是等价的。他的定理类似于20世纪80年代哈林顿-希拉和库宁的定理:$L(\mathbf {R})$对ccc强迫扩展理论的绝对性与弱紧致基数是等价的。与Itay Neeman一起,我们改进了Schindler定理,证明了$\sigma $ -闭$\ast $ - ccc集的绝对性——而不是更大的适当集的绝对性——暗示了$\aleph _1^V$在l中的显著性。这需要对证明进行根本的改变,因为Schindler的下界论证使用了Jensen的重塑强迫,尽管正确,但在这种情况下不一定是$\sigma $ -闭$\ast $ - ccc。我们的证据更像哈林顿-希拉的证据,而不是辛德勒的证据。定理6.2的证明自然地分成两个论证。在第7章中,我们将Harrington-Shelah编码实数的方法扩展到一个专门化函数中,以允许不属于l的不可数层次的树。这在定理7.4中达到高潮,它断言如果存在$X\subseteq \omega _1$和高度为$\omega _1$的树$T\subseteq \omega _1$,使得X可以沿着T编码(见定义7.3),那么ccc posets的$L(\mathbf {R})$ -绝对性必须失败。我们在第8章中完成了这个论证,在第8章中,我们证明了如果在V的任何$\sigma $ -闭扩展中没有沿树T的$X\subseteq \omega _1$可编码,那么$\aleph _1^V$在l中一定是显著的。在第9章中,我们从一个显著的基础上回顾了Schindler的一般绝对性证明,证明了该论证给出了一个逐层上界:对于与$\lambda $相关的适当集,一个强烈的$\lambda ^+$ -显著基数足以获得$L(\mathbf {R})$ -绝对性。第10章致力于部分地反转第9章的逐级上界。采用Neeman, Hierarchies of forcing公理II的方法,我们能够证明$\left |\mathbf {R}\right |\cdot \left |\lambda \right |$ -链接偏序集的$L(\mathbf {R})$ -绝对性意味着区间$[\aleph _1^V,\lambda ]$在l中是$\Sigma ^2_1$ -显著的。电子邮件:zachnorwood@gmail.com
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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