Epistemological Consequences of the Incompleteness Theorems

TL;DR

This paper explores the epistemological implications of Gödel's Incompleteness Theorems, emphasizing the inherent limitations in mechanically reproducing language semantics and proposing a new interpretation of the second theorem.

Contribution

It introduces a novel 'Metatheorem of undemonstrability of internal consistency' and clarifies the epistemological impact of the incompleteness theorems on arithmetical theories.

Findings

Non-mechanizability of truths in first-order arithmetic

Peculiarities in the model of second-order arithmetic

Correction of the common interpretation of the second incompleteness theorem

Abstract

After highlighting the cases in which the semantics of a language cannot be mechanically reproduced (in which case it is called inherent), the main epistemological consequences of the first incompleteness Theorem for the two fundamental arithmetical theories are shown: the non-mechanizability for the truths of the first-order arithmetic and the peculiarities for the model of the second-order arithmetic. Finally, the common epistemological interpretation of the second incompleteness Theorem is corrected, proposing the new "Metatheorem of undemonstrability of internal consistency".