Recursion Theory and Its Applications

在提议的项目中，蔡铭超计划研究经典递归理论，重点研究一些有趣的长期存在的问题。此外，他希望引入新的想法和概念来丰富该领域，并为不同的方法和潜在的改进调查最近的一些新结果。蔡还致力于将递归理论的应用范围扩大到应用，特别是他将继续研究可证明度，其中证明理论的结果可以使用递归理论方法，如对角化和递归定理来证明。递归理论是一个逻辑领域，研究数学实践的有效内容(例如，哪些数学过程可以由自动化机器或计算机执行)。它有经典的应用，解释了为什么一些数学问题(例如，希尔伯特第十问题)在理论上是不可解的。拟议的项目旨在通过研究新旧的公开问题来提高对递归理论的理解，并通过建立与其他领域(如证明论)的联系来扩展递归理论。拟议的项目具有潜在的应用，解释了某些算术事实如何在理论上无法证明。

In the proposed project, Cai plans to study classical recursion theory with emphasis on some interesting long-standing questions. In addition, he hopes to introduce new ideas and concepts to enrich the field, as well as to investigate some recent new results for different approaches and potential improvements. Cai also aims to expand the scope to applications of recursion theory, in particular he will continue the study of the degrees of provability, where proof-theoretic results can be proved using recursion-theoretic methods such as diagonalization and the recursion theorem.Recursion theory is a field of logic addressing effective content of mathematical practices (e.g., which of the mathematical procedures can be performed by automated machines, or computers). It has classical applications explaining why some mathematical problems (e.g., Hilbert's Tenth Problem) are theoretically unsolvable. The proposed project aims to improve the understanding of recursion theory by investigating old and new open problems, as well as to expand it by building connections to other fields such as proof theory. The proposed project has potential applications explaining how certain arithmetical facts could be theoretically unprovable.