A proof of G\"odel's incompleteness theorems using Chaitin's incompleteness theorem

TL;DR

This paper offers a novel proof of G"odel's incompleteness theorems for ZFC using Chaitin's theorem, emphasizing set-based methods and number extensions over traditional fixed-point approaches.

Contribution

It introduces a new proof technique for G"odel's theorems that relies solely on set structures and number extensions, avoiding fixed-point theorems.

Findings

Proof does not use fixed-point theorem

Unprovable statements exceed specific constructed statements

Technique highlights potential for proving more unprovability results

Abstract

G\"odel's first and second incompleteness theorems are corner stones of modern mathematics. In this article we present a new proof of these theorems for ZFC and theories containing ZFC, using Chaitin's incompleteness theorem and a very basic numbers extension. As opposed to the usual proofs, these proofs don't use any fixed-point theorem and rely solely on sets structure. Unlike in the original proof, the statements which can be shown to be unprovable by our technique exceed by far one specific statement constructed from the axiom set. Our goal is to draw attention to the technique of number extensions, which we believe can be used to prove more theorems regarding the provability and unprovability of different assertions regarding natural numbers.