Daniel Isaacson

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.

The incompleteness of formal systems for arithmetic has been a recognized fact of mathematics. The term “incompleteness” suggests that the formal system in question fails to offer a deduction which it ought to. This chapter focuses on the status of a formal system, Peano Arithmetic, and explores a viewpoint on which Peano Arithmetic occupies an intrinsic, conceptually well-defined region of arithmetical truth. The idea is that it consists of those truths which can be perceived directly from the purely arithmetical content of a categorical conceptual analysis of the notion of natural number. The chapter explores its conceptual stability in light of the apparently destabilizing effect, through extensibility, of the phenomenon of incompleteness. It also offers heuristic and conceptual support for the viewpoint that Peano Arithmetic is the strongest natural first-order system for arithmetic, and that there is a sense in which it is complete with respect to purely arithmetical truth.