This is a continuation of a previous report on an experiment in autoformalisation of Gödel's second incompleteness theorem in Agda using Claude. Using the framework built in this experiment, Claude could ``automformalise'' Chaitin's proof of the first incompleteness theorem and then the Kritchman-Raz surprise examination paradox version of the second incompleteness. As the first experiment, the project provides a case study of the strengths and limitations of current large language models in mathematics. Since Chaitin's proof involves coding programs, Claude had to represent code as ternary string and could build autonomously a parser and a continuation stack evaluation machine. The fact that we can simulate computations as expected is not completely trivial and we suggested a Gandy/Howard majorisation argument, that Claude had no problem to follow. The resulting formalisation clarifies a number of details left implicit in the original presentation and provides a fully machine-checked proof of these arguments for Church's Basic Recursive Arithmetic.

翻译：本文是前篇关于使用Claude在Agda中自动形式化哥德尔第二不完全性定理实验的延续。利用该实验构建的框架，Claude能够"自动形式化"蔡廷对第一不完全性定理的证明，进而形式化克里奇曼-拉兹"意外考试悖论"版本的第二不完全性定理。与首次实验相同，本项目为当前大型语言模型在数学领域的优势与局限性提供了案例研究。由于蔡廷的证明涉及程序编码，Claude需将代码表示为三进制字符串，并自主构建解析器与续延栈求值机。计算可按预期方式模拟这一事实并非完全平凡，我们提出的一种甘迪/霍华德主参数论证，Claude能顺利遵循。最终的形式化阐明了原始表述中若干隐含细节，并为丘奇的原始递归算术中这些论证提供了完全机器验证的证明。