Set Theory

集合论

• 批准号: 0856201
• 负责人: William Woodin
• 金额: $ 39万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-09-15 至 2013-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0856201&HistoricalAwards=false
• 关键词: Set; Theory

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). The Inner Model Program is one of thecentral areas of Set Theory. The traditional view of this program has been that of an increment program with progress measured by a progression up the hierarchy of large cardinals. It is now known that this view is not correct. A critical transition occurs at the level of exactly one supercompact cardinal, and in solving the Inner Model Program for this specific large cardinal axiom, one solves the Inner Model Program for essentially all known large cardinal axioms. The solution must necessarily yield an ultimate enlargement of Goedel's inner model L. This ultimate-L must closely approximate the parent universe within which it is constructed. The ramifications for Set Theory will be profound, ranging from new inconsistency results in ZF to combinatorial theorems of ZFC proved by exploiting the closeness of ultimate-L to V. Ultimate-L will also provide a key setting for resolving the hierarchy of large cardinals beyond the level of omega-huge cardinals which constitute essentially the strongest large cardinals axioms which are not known to be inconsistent. The point is that ultimate-L inherits large cardinals from V exactly as L inherits large cardinals from V if 0-sharp does not exist. Therefore the analysis of what is possible in ultimate-L is in effect the analysis of what is possible in V.The mathematical study of Infinity in its modern incarnation dates from thework of Cantor in the late 19th century, this area of mathematicsis Set Theory. With the results of Goedel and thenCohen from the middle of the 20th century it was established that many of thefundamental problems could not be solved on the basis of the current ZFC axioms for Set Theory. The most famous example is the problem of Cantor's ContinuumHypothesis which was placed first by Hilbert on his list of 20 questionsin 1900. The solution to this question and the numerous other questions now known to be unsolvable requires the discovery of new axioms beyond the current axioms. Over the last few years a new approach to this problem has emerged based on generalizations of Goedel's axiom of constructibility. There is now convincing evidence that this approach will lead to new axioms which settle all the problems of Set Theory currently known to be unsolvable, are compatible with all known strong axioms of infinity, and which themselves are immune to kind of unsolvable problems that are ubiquitous in Set Theory. These examples will be the first examples of such axioms ever discovered and therefore show great promise for at long last finding the correct axioms for Set Theory and thereby solving in particular the problem of the Continuum Hypothesis.

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。内模程序是集合论的核心领域之一。传统的观点认为，这一计划一直是一个增量计划的进展衡量的进展了层次结构的大枢机主教。现在知道这种看法是不正确的。一个关键的转变发生在一个超紧基数的水平上，在解决这个特定的大基数公理的内部模型程序时，人们解决了基本上所有已知的大基数公理的内部模型程序。这个解必然产生哥德尔的内部模型L的一个最终的扩大。这个终极-L必须非常接近于它所构建的母宇宙。集合论的分支将是深刻的，从ZF中的新的不一致结果到ZFC的组合定理，通过利用Ultimate-L到V的接近性来证明。Ultimate-L还将提供一个关键的设置，用于解决超出omega-巨大基数水平的大基数的层次，这些基数基本上构成了最强的大基数公理，而这些公理是不一致的。关键是，如果0-sharp不存在，则ultimate-L从V继承大基数，就像L从V继承大基数一样。因此，分析什么是可能的终极L实际上是分析什么是可能的V.数学研究的无限在其现代化身日期从thework康托在19世纪后期世纪，这方面的procticsis集理论。随着Goedel和Cohen在世纪中期的研究结果，人们认识到许多基本问题不能在当前集合论ZFC公理的基础上得到解决。最著名的例子是问题康托的ContinuumHypothesis这是第一次由希尔伯特在他的名单20个问题在1900年。要解决这个问题和许多其他现在已知无法解决的问题，需要发现超越现有公理的新公理。在过去的几年里，一个新的方法来解决这个问题已经出现的基础上推广的哥德尔公理的建设性。现在有令人信服的证据表明，这种方法将导致新的公理，解决所有的问题集理论目前已知的是不可解的，是兼容的所有已知的强公理的无穷大，并本身是免疫的种不可解的问题是无处不在的集理论。这些例子将是第一个例子，这样的公理以往任何时候都发现，因此显示出巨大的希望，在漫长的最后找到正确的公理集理论，从而解决特别是问题的连续统假设。