Mathematical Sciences: Problems in Large Cardinals, Forcing and Combinatorics

研究的领域是集合论，特别是大基数和强迫。集合论可以被视为包含数学，也就是说，经典数学的每个定理都可以被表述为关于集合的陈述，然后从集合论的标准公理集合ZFC中正式推导出来。从Godel(1936)和Cohen(1963)的工作开始，集合论者已经证明了数学家们所研究的某些问题是独立的，即从ZFC中既不能证明也不能证明。第一个也是最广为人知的例子是哥德尔-科恩的结果，即康托的连续统假设是独立的。然而，当ZFC被断言存在大的无穷大基数的公理扩充时，一些独立于ZFC的陈述仍然是可证明的。本项目包括两个部分：一是继续研究一类非常大的基数，二是研究von Neumann猜想的一致性和独立性以及一系列相关问题。这样的研究加深了我们对数学证明的局限性的理解。

The area of research is set theory, in particular, large cardinals and forcing. Set theory may be viewed as containing mathematics, that is, every theorem of classical mathematics may be formulated as a statement about sets, and then formally derived from the standard collection ZFC of axioms for set theory. Beginning with the work of Godel (1936) and Cohen (1963), set theorists have shown that certain problems, on which mathematicians have worked, are independent, i.e. neither provable nor disprovable from ZFC. The first and most well known example of this is the Godel-Cohen result that Cantor's continuum hypothesis is independent. Some statements, independent of ZFC, are nevertheless known to be provable when ZFC is augmented by axioms asserting the existence of large infinite cardinal numbers. This project has two parts: firstly, to continue the study of a class of very large cardinals, and secondly, to study the consistency and independence of von Neumann's conjecture and a collection of related problems. Such studies deepen our understanding of the limits of mathematical proof.