库尔特·哥德尔
在任何足以进行算术的一致的规则系统中，都存在该系统永远无法证明的真命题——数学真理超出了任何这样一个系统所能证明的范围。
定位
维也纳学派周围的逻辑学家：每个足以包含算术的一致形式系统，都含有它无法证明的真语句，并且无法证明自身的一致性。在哲学上，他为关于数学对象的柏拉图主义辩护。
核心思想
- 不完备性定理
- 一阶逻辑的完备性
- 数学柏拉图主义
- 旋转的宇宙
影响
受影响于 5
戈特弗里德·威廉·莱布尼茨
受其影响
Gödel studied Leibniz intensively and, like him, sought to show first that God is possible, in a modal ontological proof built on positive properties (notes 1970; publ. 1987). K. Gödel, “Ontological Proof” (1970), in Collected Works III (1995); J. H. Sobel, “Gödel’s Ontological Proof”, in J. J. Thomson (ed.), On Being and Saying (1987); SEP “Kurt Gödel”
依据: AI 草稿
戈特弗里德·威廉·莱布尼茨 › 1646–1716
格奥尔格·康托尔
受其影响
Gödel’s paper on Cantor’s continuum problem takes up Cantor’s open question and presses the iterative conception of sets. SEP “The Early Development of Set Theory” (J. Ferreirós), §§3–4 (Gödel on Cantor’s continuum problem, iterative conception)
依据: AI 草稿
格奥尔格·康托尔 › 1845–1918
埃德蒙德·胡塞尔
受其影响
Gödel studied Husserl from 1959 and proposed phenomenology as a method for clarifying the meaning of mathematical concepts and intuition (draft, 1961; publ. 1995). K. Gödel, “The Modern Development of the Foundations of Mathematics in the Light of Philosophy” (1961), in Collected Works III (1995); SEP “Kurt Gödel”
依据: AI 草稿
埃德蒙德·胡塞尔 › 1859–1938
阿尔伯特·爱因斯坦
受其影响
For Einstein’s seventieth birthday Gödel found rotating-universe solutions of his field equations and argued that relativity supports an idealistic view of time (1949). K. Gödel, “An Example of a New Type of Cosmological Solutions of Einstein’s Field Equations of Gravitation”, Reviews of Modern Physics 21 (1949); “A Remark about the Relationship between Relativity Theory and Idealistic Philosophy”, in P. A. Schilpp (ed.), Albert Einstein: Philosopher-Scientist (1949); SEP “Kurt Gödel”
依据: AI 草稿
阿尔伯特·爱因斯坦 › 1879–1955
莫里茨·石里克
受其影响
Gödel attended Schlick’s seminar on Russell’s Introduction to Mathematical Philosophy and, from 1926, meetings of the Vienna Circle, without adopting its positivism. J. W. Dawson, Logical Dilemmas: The Life and Work of Kurt Gödel (1997); SEP “Kurt Gödel”
依据: AI 草稿
莫里茨·石里克 › 1882–1936
影响了 2
鲁道夫·卡尔纳普
影响了
Carnap used Gödel’s arithmetisation to state the syntax of a language within that language and, after the incompleteness results, separated derivability from consequence (Logische Syntax der Sprache, 1934). R. Carnap, Logische Syntax der Sprache (1934); K. Gödel, “Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I”, Monatshefte für Mathematik und Physik 38 (1931); SEP “Rudolf Carnap”
依据: AI 草稿
鲁道夫·卡尔纳普 › 1891–1970
艾伦·图灵
影响了
Turing related his unsolvability proof for the Entscheidungsproblem to Gödel’s incompleteness theorem and, like Gödel’s arithmetisation, coded machines as numbers (‘On Computable Numbers’, 1936). A. M. Turing, “On Computable Numbers, with an Application to the Entscheidungsproblem”, Proceedings of the London Mathematical Society 42 (1936–37); K. Gödel, Monatshefte für Mathematik und Physik 38 (1931); SEP “Alan Turing”
依据: AI 草稿
艾伦·图灵 › 1912–1954
批评
受批评于 1
路德维希·维特根斯坦
受其批评
Wittgenstein questioned the philosophical significance of Gödel’s incompleteness theorem and doubted that ‘true but unprovable’ makes sense outside a system (Remarks on the Foundations of Mathematics, 1956). L. Wittgenstein, Bemerkungen über die Grundlagen der Mathematik (1956), part I, appendix I (3rd ed. 1978: appendix III); SEP “Wittgenstein’s Philosophy of Mathematics”
依据: AI 草稿
路德维希·维特根斯坦 › 1889–1951
批评了 2
大卫·希尔伯特 受其影响
受其影响
Gödel answered the question posed in Hilbert and Ackermann’s Grundzüge der theoretischen Logik (1928) by proving the completeness of first-order logic (dissertation 1929; publ. 1930). D. Hilbert and W. Ackermann, Grundzüge der theoretischen Logik (1928); K. Gödel, “Die Vollständigkeit der Axiome des logischen Funktionenkalküls”, Monatshefte für Mathematik und Physik 37 (1930); SEP “Kurt Gödel”
依据: AI 草稿
批评了
Gödel’s second incompleteness theorem (1931) is commonly read as defeating Hilbert’s programme of finitary consistency proofs; Gödel himself wrote that it did not contradict Hilbert’s formalist standpoint. K. Gödel, “Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I”, Monatshefte für Mathematik und Physik 38 (1931), 173–198; SEP “Hilbert’s Program”
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大卫·希尔伯特 › 1862–1943
伯特兰·罗素 受其影响
受其影响
Gödel took Principia Mathematica as the formal system of his incompleteness theorems (1931) and welcomed Russell’s early realism about classes and concepts (‘Russell’s Mathematical Logic’, 1944). K. Gödel, “Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I”, Monatshefte für Mathematik und Physik 38 (1931); “Russell’s Mathematical Logic”, in P. A. Schilpp (ed.), The Philosophy of Bertrand Russell (1944); SEP “Kurt Gödel”
依据: AI 草稿
批评了
Gödel criticised Russell’s vicious-circle principle and no-class theory as too restrictive for classical mathematics, defending classes and concepts as real objects (‘Russell’s Mathematical Logic’, 1944). K. Gödel, “Russell’s Mathematical Logic”, in P. A. Schilpp (ed.), The Philosophy of Bertrand Russell (1944), 123–153
依据: AI 草稿
伯特兰·罗素 › 1872–1970
作品
不完备性定理：这类一致的系统既不能证明所有的算术真理，也不能证明自身的一致性。
- 1947什么是康托尔的连续统问题？
数学对象独立于我们而存在，并凭借某种直觉被把握；诸如连续统假设之类悬而未决的问题仍然有一个答案。
立场
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.
生平与背景
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.
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