Godel's theorem as a corollary of impossibility of complete axiomatization of geometry

TL;DR

This paper demonstrates that Gödel's theorem implies the impossibility of fully axiomatizing all geometries, using examples of Riemannian and sigma-Riemannian geometries to illustrate the limitations of axiomatization.

Contribution

It shows that Gödel's theorem can be derived as a consequence of the inherent impossibility of complete axiomatization in geometry, highlighting limitations in formal geometric systems.

Findings

Gödel's theorem implies limitations in axiomatizing geometries.

Riemannian and sigma-Riemannian geometries exemplify partial axiomatization.

Complete axiomatization of all geometries is impossible.

Abstract

Not any geometry can be axiomatized. The paradoxical Godel's theorem starts from the supposition that any geometry can be axiomatized and goes to the result, that not any geometry can be axiomatized. One considers example of two close geometries (Riemannian geometry and $\sigma $-Riemannian one), which are constructed by different methods and distinguish in some details. The Riemannian geometry reminds such a geometry, which is only a part of the full geometry. Such a possibility is covered by the Godel's theorem.