Set Theory

该奖项根据 2009 年美国复苏和再投资法案（公法 111-5）提供资金。内模型程序是集合论的核心领域之一。该计划的传统观点是增量计划，其进展通过大基数等级的上升来衡量。现在知道，这种观点并不正确。关键转变恰好发生在一个超紧基数的级别上，并且在求解该特定大基数公理的内模型程序时，人们求解了基本上所有已知大基数公理的内模型程序。该解决方案必然会产生哥德尔内部模型 L 的最终放大。这个最终的 L 必须非常接近构建它的母宇宙。集合论的影响将是深远的，从 ZF 中新的不一致结果到通过利用终极 L 与 V 的接近性证明的 ZFC 组合定理。终极 L 还将提供一个关键设置，用于解决超出 omega-huge 基数级别的大基数层次结构，这些基数本质上构成了最强的大基数公理，而这些公理是未知的不一致。关键是终极-L 从 V 继承大基数，就像如果 0 锐利不存在，L 从 V 继承大基数一样。因此，对终极 L 中可能的分析实际上是对 V 中可能的分析。无穷大的现代化身的数学研究可以追溯到 19 世纪末康托尔的工作，这个数学领域是集合论。 20 世纪中叶，哥德尔和科恩的研究结果表明，许多基本问题无法基于当前的集合论 ZFC 公理来解决。最著名的例子是康托的连续统假设问题，该问题被希尔伯特在 1900 年列入他的 20 个问题清单中的第一位。要解决这个问题以及现在已知的许多其他无法解决的问题，需要发现当前公理之外的新公理。在过去的几年里，基于哥德尔可构造性公理的推广，出现了解决该问题的新方法。现在有令人信服的证据表明，这种方法将产生新的公理，解决目前已知的集合论无法解决的所有问题，与所有已知的无穷大公理兼容，并且它们本身不受集合论中普遍存在的无法解决的问题的影响。这些例子将是有史以来发现的此类公理的第一个例子，因此显示出最终找到集合论的正确公理并从而特别解决连续统假设问题的巨大希望。

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.