The incompleteness of an incompleteness argument

TL;DR

This paper reexamines G"odel's First Incompleteness Theorem by identifying an additional assumption in the proof and suggests that the contradiction should challenge this assumption instead of the theory's completeness.

Contribution

It offers a new perspective by isolating an extra assumption in G"odel's argument and proposes refuting this assumption rather than the completeness of formal systems.

Findings

Identifies an overlooked assumption in G"odel's proof

Reframes the contradiction as challenging the assumption, not the system's completeness

Provides a new interpretation of the Incompleteness Theorem

Abstract

G\"odel's argument for the First Incompleteness Theorem is, structurally, a proof by contradiction. This article intends to reframe the argument by, first, isolating an additional assumption the argument relies on, and then, second, arguing that the contradiction that emerges at the end should be redirected to refute this initial assumption rather than the completeness of number theory