Fulfillability, Instability, and Incompleteness
Roy T. Cook · 2022
The purpose of this essay is to publicize, and to a more limited extent, further develop, an alternate proof of Gödel’s incompleteness theorem due to Saul Kripke, based on a notion called fulfillability. Kripke’s work has been publicized in talks, but at the time of writing this essay the only published discussion of the material appears in Putnam (2000). Here, a more detailed and more accessible overview of the approach is given, centered on a novel generalization - the Instability Theorem. After setting up the technical machinery and demonstrating the Instability Theorem, we prove both the first incompleteness theorem and Lob’s theorem. We conclude with some observations regarding potential directions for future research along these lines.