Set Theory

集合论

• 批准号: 9970255
• 负责人: William Woodin
• 金额: $ 40.6万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-08-01 至 2005-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9970255&HistoricalAwards=false
• 关键词: Set; Theory

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.

小行星9970255 调查人员将继续他的研究之间的关系大基数，确定性，描述集理论，并迫使公理。 他的项目的重点是确定性的精细结构及其关于连续统假设的元数学后果。 是每一个无限集的点在线或可数，即，与整数相等，或者与线上所有点的集合相等？ 这就是康托的连续统问题。 现代集合论开始于35年前，P.J.科恩证明了康托的连续统问题在集合论的习惯公理的基础上是不可解的。 无限公理是由Goedel在50年前提出的，作为解决其他无法解决的问题的新公理的候选者。 30年的研究结果表明，这是真实的经典问题的描述集理论。 特别是，投射集理论（二阶数论的“真”公理）现在可以说是确定的。 科恩的证明的性质是这样的，这种成功不能直接重复康托的连续统问题。然而，最近的结果表明，可以取得进展，这些结果表明一个关键的不对称性的潜在解决方案的连续体问题。 这种不对称性的含义以及这一发现背后的技术是将要进行的研究的主要主题。