Choiceless set theory

无选择集合论

• 批准号: 2348371
• 负责人: Jindrich Zapletal
• 金额: $ 23.99万
• 依托单位: University of Florida
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2024
• 资助国家: 美国
• 起止时间: 2024-05-01 至 2027-04-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2348371&HistoricalAwards=false
• 关键词: Choiceless; set; theory

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.

项目研究设置了理论（ZF），而无需选择公理。选择的公理因提供对数学理论至关重要的对象而闻名，但不可能直接构建，例如自然数的超滤器。 PI最近开发了一种方法，该方法可以通过直观的复杂性来详细地对这些对象进行分层，而不管它们起源于哪个数学领域。图表的结果将是该方法组织的此类对象的广泛图表。该项目涉及研究生。该项目着手研究真实级别的无选择集理论（ZF）。选择的公理意味着具有有用的组合或代数特性的许多此类集合，例如矢量空间或田地，超流滤器或欧几里得空间的复杂分区的基础。 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，实际上是通过直观的复杂性来对它们进行分层的分层，无论其起源于它们的数学领域如何。项目的结果将是该方法组织的广泛图表，这些方法反映了NSF的构建范围和构建的依据，该奖项是由NSF的构建范围和依据的构建效果，该元素是由智力构建的，是由智力的构建依据。 标准。