Reverse Mathematics and Degrees of Unsolvability

逆向数学和不可解度

• 批准号: 0600823
• 负责人: Stephen Simpson
• 金额: $ 11万
• 依托单位: Pennsylvania State Univ University Park
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-07-01 至 2010-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0600823&HistoricalAwards=false
• 关键词: Reverse; Mathematics; Degrees; Unsolvability

Reverse mathematics is a far-reaching research program in the foundations of mathematics, wherein core mathematical theorems are classified up to logical equivalence according to which set-existence axioms are needed to prove them. The degrees of unsolvability are a well known algebraic structure arising from basic notions of relative computability and algorithmic unsolvability. In the current research project, some outstanding problems in the reverse mathematics of measure theory are being addressed in unexpected ways, by applying recent advances in algorithmic randomness and Kolmogorov complexity. In the realm of general topology, reverse mathematics itself is being extended far beyond the standard so-called "big five" equivalence classes, to encompass much stronger systems. Remarkable relationships are being revealed between, on the one hand, reverse mathematics and other foundationally interesting topics, and on the other hand, degrees of unsolvability of mass problems associated to effectively closed sets of real numbers.Reverse mathematics is a far-reaching research program in the foundations of mathematics. The purpose of reverse mathematics is to elucidate the axiomatic and logical structure of mathematics as a whole. It turns out that many core mathematical theorems fall into a relatively small number of logical classes, the so-called "big five" classes. Thus one sees a remarkably orderly structure. The purpose of the current research project is to further clarify and extend this structure. Among the technical tools being deployed are algorithmic randomness, complexity theory, and degrees of unsolvability.

逆向数学是一个影响深远的研究计划， 数学基础，其中核心数学定理是 分类为逻辑等价，根据其 需要集合存在公理来证明它们。 度 不可解性是一个众所周知的代数结构， 相对可计算性和算法的基本概念 不可解性 在目前的研究项目中，一些优秀的 测度论的逆向数学中的问题 以意想不到的方式解决，通过应用最新的进展， 算法的随机性和Kolmogorov复杂度。 领域的 一般拓扑学，逆向数学本身被扩展到 除了标准的所谓“五大”等价类之外， 包含更强大的系统。 非凡的关系是 一方面，在逆向数学和 另一方面， 大规模问题的不可解决程度 反向数学是一个影响深远的研究计划， 数学的基础 逆向数学的目的是 阐明数学的公理和逻辑结构， 整体 事实证明，许多核心数学定理 分成相对较少的逻辑类，即所谓的 “大五”类。 因此，我们可以看到一个非常有序的结构。 目前研究项目的目的是进一步明确 并扩展这个结构。 技术工具包括 部署的是算法随机性，复杂性理论， 不可解性的程度。