The Antinomy of the Liar and Provability

TL;DR

This paper explores the logical paradoxes related to self-referential sentences and provability, showing limitations in how certain statements about truth and provability can be formally expressed and proven.

Contribution

It demonstrates that sentences asserting their own unprovability cannot be proven within systems, highlighting fundamental limits of formal logic regarding truth and provability.

Findings

A sentence cannot be simply labeled as 'P is not true'

In systems where Q states 'Q is not provable,' it cannot be proven whether Q is true or not

The work clarifies the logical boundaries of self-referential statements about provability

Abstract

This work evidences that a sentence cannot be denominated by P and written as P IS NOT TRUE. It demonstrates that in a system in which Q denominates the sentence Q IS NOT PROVABLE it is not provable that Q is true and not provable.