Set Theory

调查者伍丁将继续研究大基数、决定论、描述集合论和强迫公理之间的关系。他的项目的重点是关于连续体假设的确定性的精细结构及其元数学结果。直线上的每个无限点集是可数的，即与整数相等，还是与直线上所有点的集合相等？这就是康托的连续体问题。现代集合论始于35年前，P·J·科恩证明，基于集合论的惯常公理，康托的连续统问题是不可解的。无限公理是歌德尔50年前提出的，作为解决原本无法解决的问题的新公理的候选者。30年的研究结果表明，描述集合论的经典问题也是如此。特别是，射影集合论(二阶数理论的“真”公理)现在可以说已经确定了。科恩证明的性质是，这种成功不能直接复制到康托的连续统问题上。然而，最近的结果表明可以取得进展，因为这些结果表明关于连续统问题的潜在解决方案存在关键的不对称性。这种不对称性的影响，以及这一发现背后的技术，是将要进行的研究的主要主题。

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.