Reverse Mathematics and Degrees of Unsolvability

逆数学是数学基础中一个影响深远的研究领域，它将核心的数学定理划分为逻辑等价，根据逻辑等价需要集存在公理来证明。不可解度是由相对可计算性和算法不可解性的基本概念产生的一个众所周知的代数结构。在当前的研究项目中，通过应用算法随机性和Kolmogorov复杂性的最新进展，以意想不到的方式解决了度量理论逆向数学中的一些突出问题。在一般拓扑学领域，反向数学本身正在被扩展，远远超出标准的所谓“五大”等价类，以涵盖更强大的系统。一方面，反向数学和其他有趣的基础主题之间存在着显著的关系，另一方面，与有效封闭实数集相关的质量问题的不可解程度之间存在着显著的关系。逆向数学是数学基础中影响深远的研究课题。逆向数学的目的是阐明数学作为一个整体的公理和逻辑结构。事实证明，许多核心数学定理都属于相对较少的逻辑类，即所谓的“五大”类。因此，人们看到了一个非常有序的结构。当前研究项目的目的是进一步阐明和扩展这一结构。正在部署的技术工具包括算法随机性、复杂性理论和不可解度。

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.