Some Infinitary Paradoxes and Undecidable Sentences in Peano Arithmetic

TL;DR

This paper explores infinitary paradoxes and demonstrates how they can be translated into undecidable sentences within Peano arithmetic, providing insights into the nature of incompleteness and undecidability.

Contribution

It introduces new infinitary paradoxes and shows their translation into undecidable sentences in Peano arithmetic, partly confirming G"odel's claim about paradoxes and incompleteness.

Findings

Infinitary paradoxes can be translated into undecidable sentences in Peano arithmetic.

The results support G"odel's assertion about the connection between paradoxes and incompleteness.

The paper provides new examples of undecidable statements derived from paradoxical reasoning.

Abstract

According to Chaitin, G\"odel once told him "it doesn't matter which paradox you use [to prove the First Incompleteness Theorem]". In this paper I will present a few infinitary paradoxes and show how to "translate" them to some undecidable sentences in Peano arithmetic, like what G\"odel did to the Liar paradox. The results partly verify G\"odel's claim.