A Case Study in Formalizing Hilbert's Foundations of Geometry in Coq: Establishing Key Properties of Lines in Hilbert's Axiom System

Hilbert's “Foundations of Geometry” represents a pivotal milestone in the history of mathematics. The Theorem 8 in “Foundations of Geometry” plays a crucial role by establishing a fundamental property of lines in the plane: the partitioning of the plane into two distinct regions by a line. This property is directly or indirectly utilized in numerous theorem proofs and provides a definition for determining two points either share the same side or occupy opposite sides of a specified line. This definition is essential to the description of Hilbert's axioms. Additionally, this theorem can also be regarded as a special form of the Jordan curve theorem in Topology. This paper outlines the process of formally verifying Hilbert's Theorem 8 using Coq, a reliable and rigorous proof assistant. Departing from Hilbert's original definition, we propose an alternative description of two points lying on a common side of a line and rigorously establish its equivalence to the Hilbert's definition. Building upon this equivalent definition, we completely provide a rigorous demonstration establishing the validity of the theorem. The introduction of this alternative definition enhances the conciseness of our formalization. Our work not only emphasizes the importance of Theorem 8 within Hilbert's axiom system but also lays the foundation for further exploration and verification of propositions derived from this system.