Gödelian sentences, Rosserian sentences and truth

Abstract

Abstract There is a long-standing debate in the logico-philosophical community as to if/why the Gödelian sentences of a consistent and sufficiently strong theory are true ($\gamma $ is a Gödelian sentence of $T$ when $\gamma $ is equivalent to the $T$-unprovability of $\gamma $ inside $T$). The prevalent argument seems to be something like the following: since every one of the Gödelian sentences of such a theory is equivalent to the consistency statement of the theory, even provably so inside the theory, the truth of those sentences follows from the consistency of the theory in question; so, Gödelian sentences of consistent and sufficiently strong theories are true. In this paper, we critically examine this argument and present necessary and sufficient conditions for the truth of Gödelian sentences (and Rosserian sentences) of consistent and sufficiently strong arithmetical theories.