I give an analysis of Dummett’s interpretation of the philosophical significance of the incompleteness theorem and its possible application to a rejection of the law of the excluded middle ( \(\textsf{LEM}\) ). According to Dummett, Gödel’s results question the general specifiability of a ‘principle for recognizing something true about the natural numbers’. He takes the incompleteness theorems to show that our collection of such principles is indefinitely extensible, and therefore warrant a rejection of \(\textsf{LEM}\) . First, I argue that for this claim to be successful, Dummett needs to understand general specifiability as ‘being part of a recursively enumerable collection of arithmetic truths’. I then provide a formal framework to model this notion’s indefinite extensibility. To do so, I apply potentialist ideas to extensions of theories (rather than to extensions domains) and test whether a suitable formulation of \(\textsf{LEM}\) holds in the resulting framework.
