CAREER: Large Cardinals

PI正在研究Woodin基数区域的大基数理论的几个方面，包括（1）长博弈;（2）迭代性;（3）与描述集理论的联系。1-PI通过以前的NSF支持获得的结果证明了Woodin基数和可变可数长度游戏之间的紧密联系。对于特定的大基数，存在合适的模型可以用来产生特定长度的博弈的确定性，随着长度的增加，对应于大基数强度的增加。这些游戏反过来可以用来决定声明最小内部模型的largecardinals。PI正试图扩展这种对应关系，第一个主要测试案例是长度为Ω 1的开放游戏。2-对Woodin基数的研究自然会导致一个超限博弈，称为迭代博弈。在这个博弈中，好的玩家的获胜策略（"迭代策略"）对于内在模型理论的核心比较过程是必不可少的。证明迭代策略是该领域中最重要的问题。PI一直致力于解决这个问题，为现有的部分结果做出了贡献。作为本项目的一部分，我们研究了与可迭代性的一般问题相关的几个问题，但这些问题更专业，更具体。这项研究应有助于加深对主要问题的理解，并有望导致额外的部分结果。3-PI正在努力巩固大基数和真实的数的可定义集合之间的联系。大量的工作，其中大部分是在70年代和80年代完成的，提供了一个详细的分析，可定义的集合的reals assumingdeterminacy。后来的工作获得确定性从大枢机主教。PI致力于模仿现有的可定义实数集的分析，直接从大型基数开始工作。这里的重点是试图将用于研究实数的可定义集合的方法转换为可用于内模理论和大基数研究的方法。集合论是数学的一个分支，它试图理解数学的宇宙，即数学家研究的所有对象的集合，或者更准确地说，所有集合的集合。（数字、组、函数等，都可以表示为集合）。 集合论者认为这个宇宙本身就是一个结构，这个结构本身可以用数学推理来分析。例如，一个集合论者可能会问："宇宙是否嵌入到一个类似但不完全相同的结构中？集合理论家处理这种嵌入（"基本嵌入"）的方式与处理数字上的函数的方式大致相同，他们会问"目标结构与原始结构有多相似？" "和"什么是最小的大小集实际上移动的嵌入？关于宇宙嵌入的问题形成了集合论的一个子领域，称为大基数的研究。这里的术语暗示了最后一个问题的答案。事实上，初等嵌入实际上移动的集合比其他数学分支中研究的任何对象都要大得多（例如，比真实的数的集合要大得多）。然而事实证明，初等嵌入的存在对较低层次的对象有具体的影响，包括对真实的数的具体影响。这个项目是正在进行的努力的一部分，以更好地了解大基数，更好地了解他们的影响线的实数。PI，就像这个领域的其他研究人员一样，是由抽象和具体之间的联系所激励的。

The PI is investigating several aspects of large cardinal theory in theregion of Woodin cardinals, including (1) Long Games; (2) Iterability; and(3) connections with Descriptive Set Theory. 1 - Results obtained by thePI through previous NSF support demonstrate the tight connection betweenWoodin cardinals and games of variable countable length. The existence ofiterable models for specific large cardinals can be used to yield thedeterminacy of games of specific lengths, with increasing lengthcorresponding to increasing large cardinal strength. These games in turncan be used to decide statements over minimal inner models for largecardinals. The PI is attempting to extend this correspondence, the firstmain test case being open games of length omega one. 2 - The study ofWoodin cardinals leads naturally to a transfinite game known as theiteration game. Winning strategies ("iteration strategies") for the goodplayer in this game are essential to the comparison process that lies atthe heart of inner model theory. Proving that iteration strategies existis the single most important problem in the field. The PI has been workingon this problem, contributing to the existing pool of partial results.Several problems, related to the general problem of iterability butspecialized and more concrete, are investigated as part of this project.This investigation should provide increased understanding of the mainproblem, and hopefully lead to additional partial results. 3 - The PI isworking to solidify the connections between large cardinals and definablesets of real numbers. Extensive work, much of it done during the '70s and'80s, provides a detailed analysis of definable sets of reals assumingdeterminacy. Later work obtained determinacy from large cardinals. The PIis working to emulate the existing analysis of definable sets of reals,working directly from large cardinals. The point here is to try to convertmethods used in the study of definable sets of reals, into methods whichcan be used in inner model theory and the study of large cardinals.Set Theory is a branch of Mathematics which attempts to understand theuniverse of Mathematics, that is the collection of all objects studied byMathematicians, or more precisely the collection of all _sets_. (Numbers,groups, functions, etc., can all be represented as sets.) Set Theoristsview this universe as a structure in its own right, a structure which canitself be analyzed using mathematical reasoning. For example, a SetTheorist may ask "is there an embedding of the universe into a similar,yet not identical, structure?" Set Theorists work with such embeddings("elementary embeddings") in much the same way that one would work withfunctions on numbers, asking "how similar is the target structure to theoriginal structure?" and "what's the smallest size of a set actually movedby the embedding?" Such questions about embeddings of the universe form asubfield of Set Theory known as the study of large cardinals. Theterminology here hints at the answer to the last question mentioned.Indeed, the sets actually moved by elementary embeddings are substantiallylarger than any object studied in other branches of Mathematics(substantially larger than the set of real numbers for example). Yet itturns out that the existence of elementary embeddings has concrete effectson lower level objects, including concrete effects on real numbers. Thisproject is part of the on-going effort to better understand largecardinals, and better understand their effects on the line of realnumbers. The PI, like other researchers in the field, is motivated by theconnection between the abstract and the concrete.