Formalization of Dedekind Fundamental Theorem in Coq

The completeness of real numbers is a fundamental property in real analysis, serving as the cornerstone for the development of calculus and mathematical analysis, and finding wide-ranging applications. In recent years, the significance of mathematical formalization has been consistently on the rise in the fields of mathematics and computer science, with particular emphasis on the formalization of foundational mathematical theories. The Dedekind fundamental theorem represents one form of describing the completeness of real numbers, and its formalization holds significant importance. In this paper, we employ the interactive theorem proving tool Coq, based on the Morse-Kelley axiomatic set theory, following the methodology proposed by Landau in “Foundations of Analysis” to construct the real number system and accomplish the formalized proof of Dedekind fundamental theorem. The work presented in this paper holds crucial significance in the fields of analysis and related mathematical disciplines. It lays the foundation for constructing complex number systems and formalizing theories in analysis, topology, and geometry in Coq.