Notre Dame Journal of Formal Logic

Nonstandard Models and Kripke's Proof of the Gödel Theorem

Hilary Putnam


This lecture, given at Beijing University in 1984, presents a remarkable (previously unpublished) proof of the Gödel Incompleteness Theorem due to Kripke.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 even require 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 is not at all "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).

Article information

Notre Dame J. Formal Logic, Volume 41, Number 1 (2000), 53-58.

First available in Project Euclid: 29 July 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03
Secondary: 03C62: Models of arithmetic and set theory [See also 03Hxx] 03H15: Nonstandard models of arithmetic [See also 11U10, 12L15, 13L05]

models of arithmetic nonstandard models of arithmetic


Putnam, Hilary. Nonstandard Models and Kripke's Proof of the Gödel Theorem. Notre Dame J. Formal Logic 41 (2000), no. 1, 53--58. doi:10.1305/ndjfl/1027953483.

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  • Paris, J., and L. Harrington, "A mathematical incompleteness in P"eano Arithmetic, pp. 1133–42 in Handbook of Mathematical Logic, edited by J. Barwise, North-Holland Publishing Co., 1977.