Research in Set Theory

哈姆金斯教授将从事数理逻辑领域的研究，即所谓的集合论，从事几个似乎已经成熟、有望取得进展的项目。首先，通常被认为与集合论有所区别的算术模型理论，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。斯科特的最新进展是什么？例如，这个问题涉及到算术模型和适当强迫公理的复杂混合技术。其次，大基数不可摧毁性位于强迫和大基数的交叉点，这是当代集合论研究的两个核心问题，也是Hamkins？先前的工作和最近的进展已经发现了一个相对较小的大型红衣主教的惊人的新现象。强不可折叠基数最近尤其在各种大型基数现象中作为超紧基数的令人惊讶的有效替代品，包括不可摧毁性和适当强制公理片段的一致性。第三，Hamkins教授将研究新出现的集论中关于集论宇宙的二阶和高阶特征的问题。数学逻辑和集合论的研究集中在数学基础的主题上，探索数学无限的本质和替代数学宇宙的可能性。我们对数学无限大的理解，几个世纪以来一直吸引着数学家和哲学家，现在已经在大基数层次中具体化了，Hamkins教授研究的一个中心问题将是调查大基数是如何受到强迫的影响的，强迫是Paul Cohen发明的一种技术，集合理论家通过它来构建替代的数学宇宙。这些宇宙的多样性是惊人的，集合理论家现在能够构建集合理论的模型来展示精确的预先选择的特征。在他的期末项目中，哈姆金斯教授将继续研究旨在理解宇宙和这些另类数学世界之间最基本的关系。

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.