Choiceless set theory

该项目研究集合论(ZF)，没有选择公理。众所周知，选择公理提供了对数学理论至关重要但不可能直接构造的对象，例如关于自然数的超滤子。PI最近开发了一种方法，可以通过直观的复杂性详细地对这些对象进行分层，无论它们来自哪个数学领域。图表的结果将是通过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。这个项目开始在实数集的水平上研究无选择集合论(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，例如用于向量空间或场的基、超滤子或欧氏空间的复杂划分。PI最近开发了一种方法，使证明关于此类集合的存在的详细ZF独立性结果成为可能，实际上是通过直观的复杂性对它们进行分层，而不管它们来自哪个数学领域。该项目的结果将是通过这种方法组织的此类对象的广泛图表。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.