Set Theory

该奖项是根据2009年《美国复苏和再投资法案》(公法111-5)提供资金的。内模计划是集合论的核心领域之一。这个方案的传统观点一直是一个增量方案，其进展是通过大基数等级的递增来衡量的。现在已经知道，这种观点是不正确的。一个关键的转变恰好发生在一个超紧基数的水平上，在求解这个特定的大基数公理的内模程序时，基本上解决了几乎所有已知的大基数公理的内模程序。这个解决方案必然会导致歌德尔的内部模型L的最终放大。这个最终的L必须非常接近于构建它的母宇宙。集合论的影响将是深远的，从ZF中新的不一致性结果到ZFC的组合定理，通过利用终极-L到V的贴近性来证明-终极-L还将提供一个关键设置，用于解决超出欧米伽-巨基数水平的大基数的层次结构，这些大基数公理基本上构成了已知不一致的最强大基数公理。关键是，终极-L继承了V的大基数，就像如果0-夏普不存在，L继承了V的大基数一样。因此，对终极可能的分析-L实际上就是对V的可能的分析。对无穷的现代化身的数学研究可以追溯到19世纪末康托的工作，这是数学集合论的一个领域。20世纪中叶Goedel和Then Cohen的结果表明，许多基本问题不能用目前集合论的ZFC公理来解决。最著名的例子是康托的连续假设问题，这是希尔伯特在1900年他的20个问题清单上排在第一位的问题。要解决这个问题和许多其他现在已知无法解决的问题，就需要在现有公理之外发现新的公理。在过去的几年里，基于歌德尔的可构造性公理的推广，出现了一种解决这个问题的新方法。现在有令人信服的证据表明，这种方法将导致新的公理，这些公理解决了集合论目前已知的所有不可解的问题，与所有已知的强大无穷公理兼容，并且本身不受集合论中普遍存在的那种不可解问题的影响。这些例子将是迄今发现的此类公理的第一个例子，因此显示出最终为集合论找到正确公理的巨大希望，从而特别是解决了连续统假设的问题。

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.