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《美国复苏与再投资法》（公法111-5）资助的。内部模型程序是集合理论的范围领域之一。该计划的传统观点是一个增量程序的观点，其进步是通过大型枢机主教的层次结构来衡量的。现在知道这种观点是不正确的。临界过渡发生在一个恰好的一个超级紧密基础的基本水平上，在解决此特定大型基本公理的内部模型程序时，一个人求解了本质上所有已知的大型基本公理的内部模型程序。该解决方案必须必须对Goedel的内部模型L产生最终的扩大。该最终L必须密切近似构造其构造的母体宇宙。 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.关键是，如果不存在0-sharp，则ultimate-l从V中继承了大型红衣主教。因此，对Ultimate-L中可能的分析实际上是对V中可能的分析的分析。无穷大的现代化身数学研究的历史可以追溯到19世纪后期Cantor的工作，即这一数学的这一领域。从20世纪中叶开始的Goedel和Thencohen的结果，人们确定，无法根据当前的ZFC公理来解决集合理论的许多问题。最著名的例子是康托尔（Cantor）的连续性问题的问题，希尔伯特（Hilbert）在1900年的20个问题列表中排名第一。解决这个问题的解决方案以及现在已知无法解决的其他众多其他问题，需要发现超出当前公理以外的新公理。在过去的几年中，基于Goedel的构造性公理的概括，出现了一种新的方法。现在有令人信服的证据表明，这种方法将导致新的公理，这些公理解决了当前已知的所有问题的所有问题，与所有已知的强大无穷大公理兼容，并且本身对某种无法解决的问题免疫了，这些问题在集合理论中无处不在。这些示例将是有史以来发现的这样的公理的第一个例子，因此在最后找到合适的理论公理，从而尤其解决了连续假设的问题。