Yablo's Paradox And Arithmetical Incompleteness

TL;DR

This paper explores arithmetizations of Yablo's Paradox, demonstrating their undecidability and connections to G"odel's and Jeroslow's sentences, and shows a related provable sentence using L"ob's theorem.

Contribution

It introduces two new arithmetizations of Yablo's sentences and analyzes their undecidability and relation to classical incompleteness results.

Findings

Both arithmetizations are undecidable.

Connections to G"odel and Jeroslow sentences established.

A provable related sentence is constructed using L"ob's theorem.

Abstract

In this short paper, I present a few theorems on sentences of arithmetic which are related to Yablo's Paradox as G\"odel's first undecidable sentence was related to the Liar paradox. In particular, I consider two different arithemetizations of Yablo's sentences: one resembling G\"odel's arithmetization of the Liar, with the negation outside of the provability predicate, one resembling Jeroslow's undecidable sentence, with negation inside. Both kinds of arithmetized Yablo sentence are undecidable, and connected to the consistency sentence for the ambient formal system in roughly the same manner as G\"odel and Jeroslow's sentences. Finally, I consider a sentence which is related to the Henkin sentence "I am provable" in the same way that first two arithmetizations are related to G\"odel and Jeroslow's sentences. I show that this sentence is provable, using L\"ob's theorem, as in the standard proof of the Henkin sentence.