Set Theory

9970255伍丁研究者将继续他的研究大基数、确定性、描述性集合论和强制公理之间的关系。他的项目重点是确定性的精细结构及其关于连续统假设的元数学结果。直线上的每一个无限点的集合是可数的，即与整数相等，还是与直线上所有点的集合相等？这是康托的连续统问题。现代集合论始于35年前P.J. Cohen在集合论的习惯公理基础上证明了Cantor的连续统问题是不可解的。无限公理是50年前由哥德尔提出的，作为解决其他无法解决问题的新公理的候选公理。30年的研究结果表明，这对于描述集理论的经典问题是正确的。特别是，射影集的理论（二阶数论的“真”公理）现在可以说是解决了。科恩证明的本质是这样的，这一成功不能直接用于康托尔的连续问题。然而，最近的结果表明可以取得进展，因为这些结果表明了连续统问题的潜在解决方案的关键不对称性。这种不对称的含义，以及这一发现背后的技术，是研究的主要主题。

9970255Woodin The investigator will continue his study of the relationships between large cardinals, determinacy, descriptive set theory, and forcing axioms. The focus of his project is the fine structure of determinacy and its metamathematical consequences concerning the Continuum Hypothesis. Is every infinite set of points in the line either countable, i.e., equinumerous with the integers, or else equinumerous with the set of all points in the line? This is Cantor's continuum problem. Modern set theory began 35 years ago with P.J. Cohen's proof that Cantor's continuum problem was unsolvable on the basis of the customary axioms of set theory. Axioms of Infinity were proposed by Goedel 50 years ago as candidates for new axioms that solve otherwise unsolvable problems. The results of 30 years of research have domonstrated that this is true for the classical problems of descriptive set theory. In particular, the theory of the projective sets (the ``true'' axioms for second order number theory) is now arguably settled. The nature of Cohen's proof is such that this success cannot be directly repeated for Cantor's continuum problem. Nevertheless, recent results indicate progress can be achieved, for these results suggest a key asymmetry concerning potential solutions to the continuum problem. The implications of this asymmetry, and the technology behind this discovery, are the main topics for the research to be undertaken.