Forcing and large cardinals

强迫和大基数

• 批准号: 1363364
• 负责人: Itay Neeman
• 金额: $ 42.8万
• 依托单位: University of California-Los Angeles
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-07-01 至 2019-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1363364&HistoricalAwards=false
• 关键词: Forcing; large; cardinals

The overall goal of this project is to develop a better understanding of the possible behaviors of the mathematical universe. Our knowledge of the mathematical universe comes through deduction from axioms. This knowledge is inherently incomplete, and there is a wide range of questions that cannot be decided from the standard axioms. Set theorists have developed and studied additional axioms that allow settling some of these questions. Some of these axioms are purposely applicable outside set theory; others are of a nature that is, at face value, largely internal to set theory, but turn out to have effects on basic mathematical objects, for example on sets of real numbers. This project deals with axioms of both types, and with methods that compare their relative strengths. It involves the development of new axioms of the first type that should have applications in contexts that were previously out of reach, the construction of minimal models for axioms of the second type within set theory, and applications of both the axioms and their minimal models, within set theory and to the real numbers.This project deals with several central areas in set theory: (i) forcing axioms and their applications; (ii) inner models theory; (iii) applications of inner models theory to descriptive set theory; and (iv) infinitary combinatorics. Forcing axioms are strengthenings of the Baire category theorem that allow meeting a prescribed number of dense sets with filters in prescribed classes of partial orders. In connection with (i) this project is particularly concerned with higher analogues of the proper forcing axiom (PFA). PFA, developed in the early 1980s, allows meeting $\aleph_1$ dense sets in proper partial orders. It has proved incredibly useful both as a starting point for consistency proofs and as an axiom leading to set theoretic structure theorems. Recent work of the PI shows that there are analogues of PFA which involve meeting more than $\aleph_1$ dense sets. It is one of the goals of this project to develop these analogues further, and to use them in extending applications of PFA to new contexts. The inner models program has as its main goal the construction of models for large cardinal axioms from assumptions that do not directly involve large cardinals (for example from forcing axioms). In connection with (ii), this project is primarily concerned with the construction, nature, and combinatorial properties of inner models at the level of supercompact cardinals. This is a long-standing project in the area and one that saw a great deal of recent progress. In connections with (iii) this project is concerned with applications of inner models theory at the level of Woodin cardinals to questions in descriptive set theory. The structure of inner models at this level is well understood, and there are well known connections to descriptive set theory. These connections already yielded solutions to several previously intractable questions in descriptive set theory. Finally, in connection with (iv) this project is primarily concerned with the tree property, a remnant of large cardinal strength that can consistently hold at small cardinals.

该项目的总体目标是更好地理解数学宇宙的可能行为。我们对数学宇宙的了解是通过从公理中扣除的。这些知识本质上是不完整的，并且从标准公理中无法确定许多问题。设定理论家已经开发并研究了其他公理，可以解决其中一些问题。这些公理中的一些是故意适用的外部理论。其他具有性质的性质，从表面上看，这在很大程度上是设置理论的内部，但事实证明对基本数学对象有影响，例如对实数集。该项目涉及两种类型的公理，以及比较其相对优势的方法。它涉及第一种类型的新公理的开发，该公理应该在以前无法触及的上下文中具有应用程序，构建集合理论中第二类的公理的最小模型，以及公理及其最小模型的应用，在集合理论和实数中。该项目涉及集合理论中的几个中心领域：（i）强迫公理及其应用； （ii）内部模型理论； （iii）内部模型理论在描述性集理论中的应用； （iv）无限制组合。强迫公理是Baire类别定理的增强，它允许在规定的部分订单类别中使用有过滤器来满足规定数量的密集集。与（i）有关，该项目特别与适当强迫公理（PFA）的更高类似物有关。 PFA于1980年代初期开发，允许在适当的部分订单中遇到$ \ aleph_1 $密集的套件。事实证明，这是一致性证明的起点，也是导致设定理论结构定理的公理的起点。 PI的最新工作表明，PFA的类似物涉及满足$ \ aleph_1 $密度的套件。这是该项目进一步开发这些类似物并将其用于将PFA应用于新环境的应用程序的目标之一。内部模型程序作为其主要目标是从不直接涉及大型红衣主教的假设（例如强迫公理）的假设中建造大型基本公理的模型。与（ii）相关，该项目主要与超级紧密枢机主教水平上内部模型的构建，性质和组合性质有关。这是该地区的一个长期项目，并且看到了最近的许多进展。在与（iii）的联系中，该项目涉及内部模型理论在伍丁红衣主教在描述性集理论中的问题上的应用。该级别的内部模型的结构已被充分理解，并且与描述性集理论有众所周知的联系。这些连接已经为描述性集理论中的几个以前棘手的问题提供了解决方案。最后，与（iv）有关，该项目主要与树木特性有关，这是一个可以始终如一地保持小枢机主教的大型基本强度的残余。