An implication of G\"odel's incompleteness theorem

TL;DR

This paper presents a new proof of G"odel's incompleteness theorem, explores its transfinite extension, and discusses the contradiction arising from assuming ZFC at both meta and object levels, highlighting issues with G"odel numbering.

Contribution

It introduces a novel proof of G"odel's theorem and examines the implications of assuming ZFC at both meta and object levels, revealing a fundamental contradiction.

Findings

A new proof of G"odel's incompleteness theorem is provided.

A transfinite extension of G"odel's theorem is considered.

Assuming ZFC at both levels leads to a contradiction.

Abstract

A proof of G\"odel's incompleteness theorem is given. With this new proof a transfinite extension of G\"odel's theorem is considered. It is shown that if one assumes the set theory ZFC on the meta level as well as on the object level, a contradiction arises. The cause is shown to be the implicit identification of the meta level and the object level hidden behind the G\"odel numbering. An implication of these considerations is stated.