THE INCOMPLETENESS THEOREMS: A PERSPECTIVE FROM THE FOUNDATIONS OF MATHEMATICS YONG CHENG Abstract. This is an English version of my paper from Chapter 14 of the collected volume, A Research Guide to Contemporary Philosophy of Logic (in Chinese), edited by Bo Chen. The volume is forthcoming in the National Social Science Fund Collection from People’s Publishing House, China. Published in 1931, G¨odel’s incompleteness theorems are among the most sig- nificant and profound results in the foundations of mathematics. They have had a substantial impact on the development of logic, philosophy, mathematics, the- oretical computer science, and various other fields. Drawing on a review of the latest research in the literature, this chapter summarizes and elucidates the find- ings related to the incompleteness theorems from G¨odel’s time to the present. It focuses on several key aspects: interpretations of the incompleteness theorems, their influence across different disciplines, various proofs of the theorems, gener- alizations of the theorems, and the limitations of their applicability. This article was written for general audiences to commemorate the 90th anniversary of the publication of G¨odel’s incompleteness theorems. 1. Introduction G¨ odel’s incompleteness theorems are among the most significant and profound results in the foundations of mathematics in the 20th century. They represent a milestone in the history of modern logic and have had a broad, deep, and lasting impact on the development of logic, philosophy, mathematics, theoretical computer science, and other fields, fundamentally altering the landscape of research in the foundations of mathematics since 1931. 1 1 For classic texts on the incompleteness theorems, see: (1) Herbert B. Enderton, A Mathe- matical Introduction to Logic (2nd ed.), Boston, MA: Academic Press, 2001; (2) Per Lindstr¨om, Aspects of Incompleteness, Lecture Notes in Logic, vol. 10, 1997; (3) Roman Murawski, Re- cursive Functions and Metamathematics: Problems of Completeness and Decidability, G¨odel’s Theorems, Springer Netherlands, 1999; (4) Raymond M. Smullyan, G¨odel’s Incompleteness The- orems, Oxford Logic Guides, Oxford University Press, 1992; (5) Peter Smith, An Introduction to G¨odel’s Theorems, Cambridge University Press, 2007. For research articles on the incomplete- ness theorems, see: (1) Lev D. Beklemishev, “G¨odel Incompleteness Theorems and the Limits of Their Applicability I”, Russian Math Surveys, 2010; (2) Yong Cheng, “Current Research on G¨odel’s Incompleteness Theorems”, Bulletin of Symbolic Logic, vol. 27, no. 2, 2021, pp. 113- 167; (3) Henryk Kotlarski, “The Incompleteness Theorems after 70 Years”, Annals of Pure and Applied Logic, vol. 126, no. 1-3, 2004, pp. 125-138; (4) C. Smory´ nski, “The Incompleteness 1