Research in Set Theory

Hamkins教授将在被称为集合论的数理逻辑领域进行研究，从事几个似乎已经成熟的项目。首先，理论模型的算术，通常被认为是站在有点远离集理论，有几个基本问题表现出深刻的集理论的性质，和一个跨专业的方法，现在似乎呼吁。斯科特的最新进展例如，李的问题涉及到一个复杂的混合技术，从数学模型和适当的强迫公理。第二，大基数不灭性在于强迫和大基数，当代集理论研究的两个核心问题和核心领域的哈金斯教授？先前的工作和最近的进展已经发现了一个令人惊讶的强大的新现象，相对较小的大红雀。特别是强不可折叠的基数最近在各种大的基数现象中作为超紧基数的令人惊讶的有效替代品，包括不可摧毁性和固有力公理片段的一致性。第三，Hamkins教授将研究新兴的集合论中的问题，重点是集合论宇宙的二阶和高阶特征。这个数学逻辑和集合论的研究集中在数学基础的主题上，探索数学无穷大的本质和替代数学宇宙的可能性。我们对数学无穷大的理解，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结晶，Hamkins教授研究的一个中心问题是调查大基数如何受到强迫的影响，这种技术由Paul Cohen发明，集合理论家通过它构建替代数学宇宙。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，以展示精确的预选特征。在他的最后一个项目中，Hamkins教授将继续研究旨在理解宇宙和这些替代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.