Reverse Mathematics beyond the Gödel hierarchy

超越哥德尔层次的逆向数学

• 批准号: 423971947
• 负责人: Dr. Sam Sanders
• 金额: --
• 依托单位: Institut für Philosophie II
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2019
• 资助国家: 德国
• 起止时间: 2018-12-31 至 2022-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/423971947?language=en
• 关键词: Reverse; Mathematics; beyond; del; hierarchy

David Hilbert’s famous list of 23 open problems, presented at the Paris ICM in 1900, contains a number of foundational problems: e.g. Problem 2 pertains to the consistency of mathematics, i.e. the fact that no contradiction can be proved. Hilbert later developed Problem 2 into Hilbert’s program for the foundations of mathematics, but Gödel’s famous incompleteness theorems show that this program is impossible. As a positive outgrowth, Hilbert’s notion of consistency gave rise to the Gödel hierarchy, a linear order that is said to capture essentially all natural and significant logical systems. Nonetheless, together with Dag Normann, I have recently identified a significant number of basic and natural theorems of uncountable mathematics (like the Heine-Borel compactness of the unit interval) that fall outside of the Gödel hierarchy. The aim of this project is to obtain a large collection of theorems outside of the Gödel hierarchy that form a parallel hierarchy. To this end, I will develop the following topics in Reverse Mathematics, which is a foundational program that seeks to identify the minimal axioms needed to prove theorems of ordinary mathematics.(T.1) The Reverse Mathematics of measure and integration theory, with a focus on the gauge integral. (T.2) The Reverse Mathematics of topology, with a focus on robust results.(T.3) New classes of theorems in Reverse Mathematics: uniformity, splittings, and disjunctions.Topics (T.1)-(T.3) will provide a large collection of highly natural theorems of mathematics outside the Gödel hierarchy. These topics split into sub-topics that naturally connect to and extend existing research in Reverse Mathematics. Finally, Kohlenbach's Higher-order Reverse Mathematics provides the most natural framework for the study of (T.1)-(T.3).

1900年，大卫·希尔伯特在巴黎国际数学大会上提出了著名的23个开放问题，其中包含了一些基础问题：例如，问题2涉及数学的一致性，即没有矛盾可以被证明的事实。希尔伯特后来将问题2发展成希尔伯特的数学基础计划，但Gödel著名的不完备定理表明这个计划是不可能的。作为一个积极的结果，希尔伯特的一致性概念产生了Gödel层次结构，这是一种线性秩序，据说基本上可以捕获所有自然和重要的逻辑系统。尽管如此，我和达格·诺曼（Dag Normann）一起，最近发现了大量不可数数学的基本和自然定理（如单位区间的Heine-Borel紧性），这些定理不属于Gödel层次结构。这个项目的目的是在Gödel层次结构之外获得大量定理，形成一个并行层次结构。为此，我将在反向数学中发展以下主题，反向数学是一个基础程序，旨在确定证明普通数学定理所需的最小公理。1)测量与积分理论的逆向数学，重点是规范积分。拓扑的逆向数学，重点是鲁棒性结果。逆向数学中定理的新类别：一致定理、分裂定理和析取定理。主题(T.1)-(T.3)将提供大量在Gödel层次结构之外的高度自然的数学定理。这些主题分成子主题，自然地连接并扩展了逆向数学的现有研究。最后，Kohlenbach的高阶逆数学为(T.1)-(T.3)的研究提供了最自然的框架。