G\"odel's Theorem and Direct Self-Reference

TL;DR

This paper demonstrates that direct self-reference can be effectively used to prove Gödel's incompleteness theorems, challenging the notion that such self-reference involves faulty circularity.

Contribution

It provides a novel proof of Gödel's theorems using direct self-reference, clarifying misconceptions about circularity in self-referential formulas.

Findings

Direct self-reference suffices for Gödel's proof

Challenges previous assumptions about circularity

Clarifies the logical foundations of incompleteness

Abstract

In his paper on the incompleteness theorems, G\"odel seemed to say that a direct way of constructing a formula that says of itself that it is unprovable might involve a faulty circularity. In this note, it is proved that 'direct' self-reference can actually be used to prove his result.