Nonstandard models and Kripke’s proof of the Gödel theorem.

*(English)*Zbl 1005.03054Summary: This lecture, given at Beijing University in 1984, presents a remarkable (previously unpublished) proof of the Gödel Incompleteness Theorem due to Kripke.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).

##### MSC:

03H15 | Nonstandard models of arithmetic |

03C62 | Models of arithmetic and set theory |

03F30 | First-order arithmetic and fragments |

11U09 | Model theory (number-theoretic aspects) |

11U10 | Nonstandard arithmetic (number-theoretic aspects) |

03-03 | History of mathematical logic and foundations |

01A60 | History of mathematics in the 20th century |