Unit 2 Summary: Pasch’s Axiom, Finite Sheaves & Formal Deduction

Analytical Synopsis of Vorlesungen über neuere Geometrie (1882), §§ 2, 8–12

1. Pasch’s Planar Separation Axiom (§ 2, *IV. Grundsatz*, p. 21): Any coplanar line avoiding vertices $A, B, C$ that enters open side $AB$ must exit through open side $AC$ or open side $BC$.

2. Exclusive Exit & Plane Separation (*10. Lehrsatz*, p. 25): Applying Grundsatz IV to sub-triangle $\triangle BDE$ proves no line can cut all three open sides of $\triangle ABC$, partitioning $\alpha \setminus a$ into two convex half-planes.

3. Ideal Sheaves Without Infinity (§§ 8–11): Replaces Euclidean “points at infinity” by spatial ray-sheaves (Strahlenbündel) constructed from finite plane intersections, and proves harmonic separation deductively via Grundsatz IV.

4. The Hypothetico-Deductive Manifesto (§ 12, p. 98): Inference must be completely independent of conceptual meanings and diagrams; any need to consult intuition exposes a missing axiom, while formal completeness makes projective duality automatic.