Kripke on Gödel Incompleteness

Journal article

This paper surveys six of Saul Kripke's highly creative ideas and results on Gödel incompleteness, from when he was an undergraduate to last publications. These include his extension of incompleteness from sentences to predicates, his model‐theoretic proof of incompleteness of arithmetic, his compelling analysis of incompleteness in terms of the heterological paradox rather than the liar paradox (cited by Gödel) for a heuristic account of incompleteness, his ingenious demonstration that Hilbert's programme bore within itself the seeds of its collapse independently of Gödel incompleteness, his revisionist point that there was incompleteness in mathematics well before the Paris‐Harrington Theorem, and his demonstration, contrary to the common view, that direct self‐reference need not be contradictory and can be used to obtain a Gödel sentence containing a numeral for its own Gödel number. Kripke published the work surveyed here with the exception of his model‐theoretic proof of incompleteness, which he presented in a lecture in 1978. These ideas constitute mathematical results of great ingenuity and highly illuminating philosophical insights.

Authors: Isaacson, D

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Wiley
• Journal: Theoria
• Article number: e70089
• Publication date: 2026-05-19
• Acceptance date: 2026-04-14
• DOI: 10.1111/theo.70089
• EISSN: 1755-2567
• ISSN: 0040-5825
• Language: English
• Keywords: fulfillability; direct self‐reference; model‐theoretic incompleteness; mathematical incompleteness; flexible predicates; collapse of Hilbert's programme; Grelling's heterological paradox; Gödel incompleteness