Set Theory

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。这个程序的传统观点是一个增量程序，其进度是通过大基数的层次结构的进展来衡量的。现在知道这种观点是不正确的。一个临界跃迁恰好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，一个人解决了基本上所有已知的大基数公理的内部模型程序。这个解必须产生哥德尔内模型l的最终扩展，这个最终模型l必须非常接近它所在的母宇宙。集合论的影响将是深远的，从ZF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也将提供一个关键的设置，用于解决大基数的层次结构，而不是ω -巨基数，这些大基数本质上构成了不一致的最强大基数公理。关键在于，如果0-sharp不存在，那么ultimate-L从V继承大基数，就像L从V继承大基数一样。因此，对终极l的可能性的分析实际上就是对v的可能性的分析。数学对无穷的现代研究可以追溯到19世纪晚期康托尔的工作，这个数学领域的集合论。随着哥德尔和科恩从20世纪中期开始的结果，人们确定了许多基本问题不能在当前的集合论ZFC公理的基础上解决。最著名的例子是康托尔的连续假设问题，1900年，希尔伯特在他的20个问题清单中把它排在第一位。要解决这个问题以及其他许多已知无法解决的问题，需要在现有公理之外发现新的公理。在过去的几年里，基于哥德尔构造性公理的推广，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解决集合论中目前已知的所有不可解的问题，与所有已知的强无穷公理兼容，并且它们本身不受集合论中普遍存在的不可解问题的影响。这些例子将是此类公理的第一个被发现的例子，因此对最终找到集合论的正确公理，从而解决特别是连续统假设的问题，显示出巨大的希望。

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.