Research in Set Theory

集合论研究

• 批准号: 0800762
• 负责人: Joel Hamkins
• 金额: $ 21万
• 依托单位: CUNY College of Staten Island
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-06-01 至 2012-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0800762&HistoricalAwards=false
• 关键词: Research; Set; Theory

Professor Hamkins will undertake research in the area of mathematical logic known as set theory, pursuing several projects that appear to be ripe for progress. First, the theory of models of arithmetic, usually considered to stand somewhat apart from set theory, has several fundamental questions exhibiting a deep set-theoretic nature, and an inter-speciality approach now seems called for. The most recent advances on Scott?s problem, for example, involve a sophisticated blend of techniques from models of arithmetic and the Proper Forcing Axiom. Second, large cardinal indestructibility lies at the intersection of forcing and large cardinals, two central concerns of contemporary set-theoretic research and the core area of much of Professor Hamkins? prior work, and recent advances have uncovered a surprisingly robust new phenomenon for relatively small large cardinals. The strongly unfoldable cardinals especially have served recently as a surprisingly efficacious substitute for supercompact cardinals in various large cardinal phenomena, including indestructibility and the consistency of fragments of the Proper Forcing Axiom. Third, Professor Hamkins will investigate questions in the emerging set-theoretic focus on second and higher order features of the set-theoretic universe.This research in mathematical logic and set theory concentrates on topics at the foundations of mathematics, exploring the nature of mathematical infinity and the possibility of alternative mathematical universes. Our understanding of mathematical infinity, fascinating mathematicians and philosophers for centuries, has now crystallized in the large cardinal hierarchy, and a central concern of Professor Hamkins' research will be to investigate how large cardinals are affected by forcing, the technique invented by Paul Cohen by which set theorists construct alternative mathematical universes. The diversity of these universes is astonishing, and set theorists are now able to construct models of set theory to exhibit precise pre-selected features.In his final project, Professor Hamkins will pursue research aimed at an understanding of the most fundamental relations between the universe and these alternative mathematical worlds.

哈姆金斯教授将在被称为集合论的数理逻辑领域进行研究，并开展几个似乎已经成熟的项目。首先，算术模型理论通常被认为与集合论有所不同，它有几个基本问​​题，表现出深刻的集合论性质，现在似乎需要一种跨专业的方法。例如，斯科特问题的最新进展涉及算术模型和适当强制公理技术的复杂结合。其次，大基数的不可毁性位于强迫基数和大基数的交叉点，这是当代集合论研究的两个核心问题，也是哈姆金斯教授大部分研究的核心领域。先前的工作和最近的进展揭示了相对较小的大基数的令人惊讶的强大新现象。特别是，强可展开基数最近在各种大型基数现象中令人惊讶地有效地替代了超紧凑基数，包括不可破坏性和适当强制公理片段的一致性。第三，哈姆金斯教授将研究新兴集合论中的问题，重点关注集合论宇宙的二阶和高阶特征。这项数学逻辑和集合论研究集中于数学基础的主题，探索数学无穷大的本质和替代数学宇宙的可能性。我们对数学无穷的理解，几个世纪以来一直让数学家和哲学家着迷，现在已经在大基数层次结构中具体化了，哈姆金斯教授研究的一个中心问题将是研究大基数如何受到强迫的影响，这是保罗·科恩发明的技术，集合论学家通过该技术构建了替代的数学宇宙。这些宇宙的多样性是惊人的，集合论学家现在能够构建集合论模型来展示精确的预先选择的特征。在他的最终项目中，哈姆金斯教授将进行旨在理解宇宙与这些替代数学世界之间最基本关系的研究。