Choiceless set theory

本课题研究不含选择公理的集合论。选择公理以提供数学理论的核心对象而闻名，这些对象是不可能直接构造的，例如自然数上的超过滤器。PI最近开发了一种方法，可以根据直观的复杂性对这些对象进行分层，而不管它们来自哪个数学领域。该图表的结果将是由这种方法组织的此类对象的广泛图表。这个项目涉及研究生。本课题在实数集的层次上研究无选择集理论（ZF）。选择公理意味着存在许多具有有用的组合或代数性质的这样的集合，例如向量空间或场的基、超滤子或欧几里得空间的复杂划分。PI最近开发了一种方法，可以证明关于这些集合存在的详细的ZF独立性结果，实际上是通过直观的复杂性将它们分层，而不管它们起源于哪个数学领域。项目的结果将是用这种方法组织的这些对象的广泛图表。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The project studies set theory (ZF) without the axiom of choice. The axiom of choice is known for providing objects which are central to mathematical theory yet are impossible to directly construct, such as ultrafilters on natural numbers. The PI recently developed a method which makes it possible to stratify such objects by intuitive complexity in detail regardless of which field of mathematics they originate. The result of the chart will be an extensive chart of such objects organized by this method. The project involves graduate students.This project sets out to study choiceless set theory (ZF) at the level of sets of reals. The axiom of choice implies the existence of numerous such sets with useful combinatorial or algebraic properties, such as bases for vector spaces or fields, ultrafilters, or complicated partitions of Euclidean spaces. The PI recently developed a method which makes it possible to prove detailed ZF independence results regarding the existence of such sets, in effect stratifying them by intuitive complexity regardless of the field of mathematics they originate in. The result of the project will be an extensive chart of such objects organized by this method.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.