कुर्ट गेडेल
अंकगणित के लिए पर्याप्त प्रबल किसी भी संगत नियम-प्रणाली में ऐसे सत्य कथन होते हैं जिन्हें वह कभी सिद्ध नहीं कर सकती – गणितीय सत्य ऐसी किसी एक प्रणाली की सिद्ध करने की क्षमता से आगे निकल जाता है।
पृष्ठभूमि
वियना सर्कल की कक्षा के तर्कशास्त्री: अंकगणित के लिए पर्याप्त समृद्ध हर संगत औपचारिक प्रणाली में ऐसे सत्य वाक्य होते हैं जिन्हें वह सिद्ध नहीं कर सकती, और वह अपनी ही संगति सिद्ध नहीं कर सकती। दर्शन में उन्होंने गणितीय वस्तुओं के बारे में प्लेटोवाद का बचाव किया।
मुख्य विचार
- अपूर्णता प्रमेय
- प्रथम-क्रम तर्कशास्त्र की पूर्णता
- गणितीय प्लेटोवाद
- घूर्णनशील ब्रह्मांड
प्रभाव
जिनसे प्रभावित 5
गॉटफ़्रीड विल्हेल्म लाइबनित्स
जिनसे प्रभावित
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”
साक्ष्य: AI मसौदा
गॉटफ़्रीड विल्हेल्म लाइबनित्स › 1646–1716
जॉर्ज कांटॉर
जिनसे प्रभावित
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)
साक्ष्य: AI मसौदा
जॉर्ज कांटॉर › 1845–1918
एडमुंड हुसल
जिनसे प्रभावित
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”
साक्ष्य: AI मसौदा
एडमुंड हुसल › 1859–1938
अल्बर्ट आइंस्टीन
जिनसे प्रभावित
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”
साक्ष्य: AI मसौदा
अल्बर्ट आइंस्टीन › 1879–1955
मारित्ज़ श्लिक
जिनसे प्रभावित
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”
साक्ष्य: AI मसौदा
मारित्ज़ श्लिक › 1882–1936
जिन्हें प्रभावित किया 2
रुडोल्फ कार्नैप
जिन्हें प्रभावित किया
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”
साक्ष्य: AI मसौदा
रुडोल्फ कार्नैप › 1891–1970
एलेन ट्यूरिंग
जिन्हें प्रभावित किया
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”
साक्ष्य: AI मसौदा
एलेन ट्यूरिंग › 1912–1954
आलोचना
जिनके द्वारा आलोचित 1
लुडविग विट्गेंश्टाइन
जिनके द्वारा आलोचित
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”
साक्ष्य: AI मसौदा
लुडविग विट्गेंश्टाइन › 1889–1951
जिनकी आलोचना की 2
डेविड हिल्बर्ट जिनसे प्रभावित
जिनसे प्रभावित
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”
साक्ष्य: AI मसौदा
जिनकी आलोचना की
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”
साक्ष्य: AI मसौदा
डेविड हिल्बर्ट › 1862–1943
बर्टराण्ड रसेल जिनसे प्रभावित
जिनसे प्रभावित
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”
साक्ष्य: AI मसौदा
जिनकी आलोचना की
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
साक्ष्य: AI मसौदा
बर्टराण्ड रसेल › 1872–1970
कृतियाँ
अपूर्णता प्रमेय: इस प्रकार की कोई संगत प्रणाली सारे अंकगणितीय सत्यों को, या अपनी ही संगति को, सिद्ध नहीं कर सकती।
गणितीय वस्तुएँ हमसे स्वतंत्र अस्तित्व रखती हैं और एक तरह के अंतर्ज्ञान से ग्रहण की जाती हैं; सातत्य परिकल्पना जैसे अनिर्णीत प्रश्नों का भी एक उत्तर होता है।
मत
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.
जीवन और संदर्भ
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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