The Interpretability of Inconsistency: Feferman's Theorem and Related Results

Abstract

This paper is an exposition of Feferman's Theorem concerning the interpretability of inconsistency and of further insights directly connected to this result. Feferman's Theorem is a strengthening of the Second Incompleteness Theorem. It says, in metaphorical paraphrase, that it is not just the case that a theory fails to prove its own consistency, but that a theory actively holds its own inconsistency for possible. We first give a careful presentation of the result. Then, we provide two versions of the result that are both modal and infinitary. We explain how Feferman's Theorem is connected with two notions of completion of a theory. We provide an example of an application of the theorem. Finally, we discuss the failure of the result in a constructive setting.