Herbrand's theorem and non-Euclidean geometry

TL;DR

This paper presents a new, concise logical proof that Euclid's parallel axiom cannot be derived from other Euclidean axioms, using Herbrand's theorem and basic geometric properties.

Contribution

It introduces a novel proof method combining Herbrand's theorem with classical geometric properties, simplifying the demonstration of the independence of the parallel axiom.

Findings

Herbrand's theorem can be applied to geometric independence proofs.

The proof is shorter and more intuitive than previous model-based approaches.

The approach bridges logic and geometry in a novel way.

Abstract

We use Herbrand's theorem to give a new proof that Euclid's parallel axiom is not derivable from the other axioms of first-order Euclidean geometry. Previous proofs involve constructing models of non-Euclidean geometry. This proof uses a very old and basic theorem of logic together with some simple properties of ruler-and-compass constructions to give a short, simple, and intuitively appealing proof.