## Notre Dame Journal of Formal Logic

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

### 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This lecture, given at Beijing University in 1984, presents a remarkable (previously unpublished) proof of the Gödel Incompleteness Theorem due to Kripke.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 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

**Source**

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

**Dates**

First available in Project Euclid: 29 July 2002

**Permanent link to this document**

http://projecteuclid.org/euclid.ndjfl/1027953483

**Digital Object Identifier**

doi:10.1305/ndjfl/1027953483

**Mathematical Reviews number (MathSciNet)**

MR1915131

**Zentralblatt MATH identifier**

1005.03054

**Subjects**

Primary: 03

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

**Keywords**

models of arithmetic nonstandard models of arithmetic

#### Citation

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. http://projecteuclid.org/euclid.ndjfl/1027953483.