Choiceless set theory

该项目研究没有选择公理的集合论（ZF）。选择公理以提供数学理论的核心但不可能直接构造的对象而闻名，例如自然数上的超滤子。PI最近开发了一种方法，可以通过直观的复杂性对这些对象进行详细的分层，而不管它们起源于哪个数学领域。图表的结果将是一个广泛的图表，这些对象组织的这种方法。该项目涉及研究生。该项目开始在实数集合的水平上研究无选择集合论（ZF）。选择公理意味着存在许多这样的集合，它们具有有用的组合或代数性质，例如向量空间或域的基、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，该方法可以证明有关此类集合存在的详细ZF独立性结果，实际上通过直观的复杂性对其进行分层，而不管它们起源于哪个数学领域。该项目的结果将是一个广泛的图表，这些对象组织的这种方法。这个奖项反映了NSF的法定使命，并已被认为是值得支持的评估使用基金会的智力价值和更广泛的影响审查标准。

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.