Reverse Mathematics and Degrees of Unsolvability

逆数学是数学基础中一个影响深远的研究项目，其中核心的数学定理被归类到逻辑等价，根据逻辑等价需要集合存在公理来证明它们。不可解度是一种众所周知的代数结构，它源于相对可计算性和算法不可解性的基本概念。在目前的研究项目中，通过应用算法随机性和Kolmogorov复杂性方面的最新进展，正在以意想不到的方式解决测度论反向数学中的一些突出问题。在一般拓扑学领域，逆数学本身正在远远超出标准的所谓的“五大”等价类，涵盖更强大的系统。一方面，逆数学与其他基础有趣的主题之间的显著关系正在被揭示，另一方面，与有效封闭的实数集相关的质量问题的不可解度之间的关系正在被揭示。逆数学是数学基础中一个影响深远的研究项目。逆数学的目的是阐明整个数学的公理和逻辑结构。事实证明，许多核心数学定理归入了相对较少的几个逻辑类，即所谓的“大五”类。因此，人们看到了一个非常有序的结构。本研究项目的目的是为了进一步澄清和延伸这一结构。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.