On the Truth of G\"odelian and Rosserian Sentences

TL;DR

This paper investigates the conditions under which G"odelian and Rosserian sentences are true, emphasizing the importance of soundness over mere consistency, and provides criteria for their truth in formal theories.

Contribution

It establishes that G"odelian sentences are true only in sound theories and offers necessary and sufficient conditions for the truth of G"odelian and Rosserian sentences.

Findings

G"odelian sentences are true in sound theories.

A hierarchy of conditions from consistency to soundness affects truth.

Necessary and sufficient conditions for the truth of Rosserian sentences.

Abstract

There is a longstanding debate in the logico-philosophical community as to why the G\"odelian sentences of a consistent and sufficiently strong theory are true. The prevalent argument seems to be something like this: since every one of the G\"odelian sentences of such a theory is equivalent to the theory's consistency statement, even provably so inside the theory, the truth of those sentences follows from the consistency of the theory in question. So, G\"odelian sentences of consistent theories should be true. In this paper, we show that G\"odelian sentences of only sound theories are true; and there is a long road from consistency to soundness, indeed a hierarchy of conditions which are satisfied by some theories and falsified by others. We also study the truth of Rosserian sentences and provide necessary and sufficient conditions for the truth of Rosserian (and also G\"odelian) sentences of theories.