Mathematical Sciences: Problems in Large Cardinals, Forcing and Combinatorics

该项目的研究领域是集合论， 特别是大基数和强迫。 集合论可能是 被视为包含数学，也就是说， 经典数学可以表述为关于 集合，然后从标准集合ZFC正式派生 集合论的公理 从哥德尔的作品开始 （1936）和科恩（1963），集合理论家已经表明，问题 数学家们所研究的是独立的，也就是说， 从ZFC可以证明也可以证明。 第一个也是最著名的 这方面的例子是哥德尔-科恩的结果，康托的连续统， 假设是独立的 一些独立于ZFC的声明 然而，当ZFC被扩充为 断言存在大无穷基数的公理 号码 该项目有三个部分：第一，继续 研究一类非常大的红衣主教;第二，继续 研究这些红衣主教的问题， 有限组合学;第三，研究一些无关的主题。 虽然在这方面的证据的细节主要是指 对于专家来说，这是一个非常有趣的发现， 非常基本的数学命题不仅还没有 证明或否定，但可以证明是不可证明的（或 可证伪的）。

The area of research of this project 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 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. The project has three parts: first, to continue the study of a class of very large cardinals; second, to continue the study of the relation of these cardinals to problems in finite combinatorics; and third, to study some unrelated topics. While the details of proofs in this area are mainly meant for experts, it is a remarkably intriguing discovery that some very basic mathematical propositions not only have not yet been proved or disproved, but can be proved to be unprovable (or disprovable).