The Grundlagen der Geometrie brought Euclid’s ancient axioms up to the standards
of modern logic, anticipating a completely mechanical verification of their theorems.
There are five groups of axioms, each focused on a logical feature of Euclidean geometry.
The first two groups give us ordered geometry, a highly limited setting where
there is no talk of measure or angle. From these, we mechanically verify the Polygonal
Jordan Curve Theorem, a result of much generality given the setting, and subtle
enough to warrant a full verification.
Along the way, we describe and implement a general-purpose algebraic language
for proof search, which we use to automate arguments from the first axiom group. We
then follow Hilbert through the preliminary definitions and theorems that lead up to
his statement of the Polygonal Jordan Curve Theorem. These, once formalised and
verified, give us a final piece of automation. Suitably armed, we can then tackle the
main theorem
