Kurt Gödel

Dzieła

• 1931O formalnie nierozstrzygalnych zdaniach Principia Mathematica i pokrewnych systemów I

  Twierdzenia o niezupełności: żaden niesprzeczny system tego rodzaju nie może udowodnić wszystkich prawd arytmetyki ani własnej niesprzeczności.

• 1947What Is Cantor’s Continuum Problem? (Czym jest problem continuum Cantora?)

  Przedmioty matematyczne istnieją niezależnie od nas i są ujmowane przez rodzaj intuicji; nierozstrzygnięte pytania, takie jak hipoteza continuum, wciąż mają odpowiedź.

Stanowisko

The paper of 1931 showed that any consistent formal system that contains elementary arithmetic and whose axioms can be listed mechanically leaves some arithmetical sentences undecided: there is a true sentence that it cannot prove. A second theorem adds that such a system cannot prove its own consistency. Gödel achieved this by coding formulas and proofs as numbers, so that a sentence could in effect say of itself that it is unprovable. The result ended the hope of Hilbert’s programme to secure all of mathematics by finite consistency proofs, and his method of arithmetisation was taken up by Turing and Carnap. Gödel also showed that the axiom of choice and the continuum hypothesis are consistent with the standard axioms of set theory (1938–40). Philosophically he was a Platonist: mathematical objects exist independently of us and are grasped by a kind of intuition, so questions like the continuum hypothesis have a definite answer even if the axioms do not decide it. He criticised Russell’s no-class theory and, in drafts never published in his lifetime, Carnap’s view of mathematics as syntax. He studied Leibniz and from the late 1950s Husserl, found solutions to Einstein’s equations with rotating universes (1949), and left notes for a modal version of the ontological proof of God.

Życie i kontekst

Kurt Gödel was born in 1906 in Brünn (Brno) into a German-speaking family. From 1924 he studied physics and then mathematics in Vienna, where he attended Moritz Schlick’s seminar and meetings of the Vienna Circle without sharing its positivism. His dissertation under Hans Hahn (1929) proved the completeness of first-order logic; the incompleteness theorems followed in 1931. He visited the Institute for Advanced Study in Princeton several times from 1933 and suffered repeated depressive crises. After the annexation of Austria his position as a university lecturer lapsed, and in 1940 he and his wife Adele emigrated via the Trans-Siberian railway and Japan. In Princeton he was a daily walking companion of Einstein. He became a US citizen in 1948. Fearing poisoning, he ate almost nothing while Adele was in hospital, and he died in Princeton in January 1978.

Dalsze główne dzieła

• The Completeness of the Axioms of the Functional Calculus of Logic (dissertation 1929, published 1930)
• On Formally Undecidable Propositions of Principia Mathematica and Related Systems I (1931)
• The Consistency of the Continuum Hypothesis (1940)
• Russell’s Mathematical Logic (1944)
• An Example of a New Type of Cosmological Solutions of Einstein’s Field Equations (1949)