On Arithmetical Truth of the Self-Referential Sentences

TL;DR

This paper generalizes Gödel's argument about self-referential sentences claiming they are unprovable, examining its validity and implications for arithmetical truth.

Contribution

It extends Gödel's original argument to a broader class of self-referential sentences and analyzes their truthfulness within formal systems.

Findings

The generalized argument remains valid under certain conditions.

Self-referential sentences asserting unprovability can be true, supporting Gödel's original conclusion.

The validity of the argument depends on specific logical assumptions.

Abstract

We take an argument of G\"odel's from his ground-breaking 1931 paper, generalize it, and examine its validity. The argument in question is this: the sentence $G$ says about itself that it is not provable, and $G$ is indeed not provable; therefore, $G$ is true.