Nonstandard Models and Kripke’s Proof of The Gödel Theorem

Abstract

This lecture, given at Beijing University in 1984, presents a remarkable (previously unpublished) proof of the Gödel Incompleteness Theorem due to Kripke. Today we know purely algebraic techniques that can be used to give direct proofs of the existence of nonstandard models in a style with which ordinary mathematicians feel perfectly comfortable–techniques that do not even require knowledge of the Completeness Theorem or evenrequire that logic itself be axiomatized. Kripke used these techniques to establish incompleteness by means that could, in principle, have been understood by nineteenth-century mathematicians. The proof exhibits a statement of number theory–one which isnot atall “self referring” –and constructs two models, in one of which it is true and in the other of which it is false, thereby establishing “undecidability” (independence). © 2000 by the University of Notre Dame. All rights reserved.