Inner Model Theory and Descriptive Set Theory

内模型理论和描述集合论

• 批准号: 0401312
• 负责人: John Steel
• 金额: $ 38万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-07-01 至 2009-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0401312&HistoricalAwards=false
• 关键词: Inner; Model; Theory; Descriptive; Set

Steel will work on the problem of constructing canonical inner modelsfor large cardinal hypotheses, and on related problems in pure descriptiveset theory. One very well known and central problem in this area iswhether the Proper Forcing Axiom is equiconsistent with the existence of asupercompact cardinal. A solution to this problem is probably still faroff, but Steel has recently made some progress using the powerful andflexible ``core model induction" method. He intends to pursue and developthis method further, as it applies to the Proper Forcing Axiom, and inseveral other contexts as well. This program leads naturally to a numberof questions concerning the relationship between canonical inner modelswith Woodin cardinals and the derived models of the Axiom of Determinacywhich are associated to them. Steel has some results on this connection,and plans to investigate the area further. Much of set theory is motivated by the simple question``What are the proper axioms for mathematics?".Mathematiciansprove things for a living; what should they take as theircommon assumptions in these proofs? In the period 1905-1927, Russell,Zermelo, Fraenkel, and Skolemisolated an elegant list of basic statements about sets, expressedin the language of set theory, and showed that from these axiomsone could derive all of the mathematics of the time. This system ofaxioms is now known as Zermelo-Fraenkel Set Theory with Choice,or ZFC. While there is still no hint of a mathematical statement whichcannot be expressed in the language of set theory, we havediscovered that ZFC is incomplete in important ways. Asurprising number of quite basic questions about sets in generalare not decided by the axioms of ZFC; moreover, many of themore abstract questions of analysis, algebra, and topology aresimilarly left undecided. This leads to Godel's Program, first formulatedby Kurt Godel in the late 1940's:Decidemathematically interesting questionswhich are independent of ZFC in well-justified extensions ofZFC. There have been great successes in this direction obtainedby adding ``large cardinal hypotheses" to ZFC.Such hypothesesassert the existence of sets whose cardinality, or size, isinaccessible from below in ever stronger senses. Possibly our deepestunderstanding of large cardinal hypothesescomes from the inner model program. This program attempts toassociate to each large cardinal hypothesis H a canonical minimaluniverse of sets (or ``inner model") in which H is true. The strongerH is, the largere this universe must be.The inner modelswe have so far constructed have internal structures whichadmit a systematic, detailed analysis, a ``fine structure theory"which gives us a very good idea as to what these universeslook like. Inner models are crucial in severalbasic uses of large cardinal hypotheses to settle questionsleft undecided by ZFC. Steel will focus his efforts on furtheringthe inner model program.

钢将工作的问题，建设规范的内部modelsfor大型基数假设，并在纯constitutivity理论的相关问题。这一领域中一个非常著名的中心问题是真力公理是否与紧基数的存在性等相容。这个问题的解决可能还很遥远，但斯蒂尔最近利用强大而灵活的"核心模型感应”方法取得了一些进展。他打算进一步追求和发展这种方法，因为它适用于适当的强制公理，并在其他几个方面。这个程序自然导致了一些问题，关于规范内部modelswith Woodin基数和派生模型的公理的决定性，这是与他们相关的关系。斯蒂尔在这方面有了一些结果，并计划进一步调查这一领域。 集合论的大部分是由一个简单的问题所激发的：“什么是数学的正确公理？”数学家以证明事物为生;在这些证明中，他们应该把什么作为他们的共同假设？在1905年至1927年期间，罗素，策梅洛，弗伦克尔和斯科伦分离出一个优雅的基本语句列表，在集合论的语言表达，并表明，从这些公理可以得出所有的数学的时间。这个公理系统现在被称为Zermelo-Fraenkel选择集理论，或ZFC。虽然仍然没有暗示一个数学陈述，不能表达在语言的集合论，我们已经发现，ZFC是不完整的重要方式。令人惊讶的是，关于集合的基本问题中，有许多不是由ZFC公理决定的;此外，许多更抽象的分析、代数和拓扑问题也同样没有决定。这导致了哥德尔的计划，首先制定了库尔特哥德尔在20世纪40年代后期：决定数学上有趣的问题，这是独立的ZFC在合理的扩展ZFC。通过在ZFC中加入“大基数假设”，在这个方向上已经取得了巨大的成功。这样的假设断言存在这样的集合，它们的基数或大小在更强的意义上是从下面不可接近的。也许我们对大基数假设最深刻的理解来自于内部模型程序。这个程序试图将每个大基数假设H与一个H为真的集合的规范极小论域（或“内部模型”）联系起来。H越强，这个宇宙就越大。我们迄今为止构建的内部模型具有内部结构，这些内部结构允许进行系统的、详细的分析，一种“精细结构理论”，它给了我们一个关于这些宇宙是什么样子的很好的想法。内部模型在大基数假设的几个基本用途中是至关重要的，以解决ZFC遗留的问题。斯蒂尔将集中精力推进内部模型计划。