Mathematical Sciences: Set Theory

数学科学：集合论

• 批准号: 9322442
• 负责人: William Woodin
• 金额: $ 27.5万
• 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: 1994
• 资助国家: 美国
• 起止时间: 1994-07-15 至 2000-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9322442&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Set; Theory

9322442 Woodin The focus of the project is the relationship between large cardinal axioms and determinacy axioms. This is interpreted somewhat broadly and is meant to include, for example, the study of a new class of models. These models are generated as forcing extensions of models in which the axiom of determinacy holds. They are interesting because they provide the first examples of canonical models in which the Continuum Hypothesis is false. One consequence of Kurt Godel's results of 60 years ago is that there will always be mathematical problems which cannot be solved. This first became a practical concern 30 years ago when Paul Cohen showed that Cantor's Continuum Hypothesis cannot be settled. Since Cohen's work, numerous problems have been shown to be unsolvable. The large cardinal hierarchy provides in many cases a resolution to the problem of independence. This is done, not by resolving the question, but rather by calibrating the degree to which it is unsolvable. This hierarchy of axioms, each of which asserts the existence of a new infinity, emerges as the fundamental mathematical reality. The investigator intends to study this hierarchy in its many manifestations. ***

9322442伍丁该项目的重点是大型基数公理和确定性公理之间的关系。这有点宽泛地解释，例如，它的意思是包括对一类新模型的研究。这些模型是作为确定性公理成立的模型的强制扩展而产生的。它们之所以有趣，是因为它们提供了规范模型的第一个例子，在这些模型中，连续统假设是错误的。60年前库尔特·戈德尔的研究结果的一个结果是，总是会有无法解决的数学问题。30年前，当保罗·科恩证明康托的连续统假说不能成立时，这第一次成为了一个实际问题。自从科恩的工作以来，无数的问题被证明是无法解决的。在许多情况下，大的基本等级制度为独立问题提供了解决办法。要做到这一点，不是通过解决问题，而是通过校准问题无法解决的程度。这种公理的层级结构，每个公理都断言一个新的无穷大的存在，成为基本的数学现实。这位研究者打算研究这种等级制度的许多表现形式。***