Kurt Gödel (April 28th 1906 - January 14th 1978) is a Mathématicien and logician.

Its result more known, the Theorem of incomplétude of Gödel, affirms that any sufficiently powerful logical system to describe the Arithmétique entireties admits proposals on the integers not being able to be neither cancelled nor confirmed starting from the Axiome S of the Théorie. Gödel also showed the complétude of the calculation of the first order predicates. It as showed the relative coherence of the Hypothèse of continuous the, showing as it cannot be refuted starting from the allowed axioms of the Set theory, by admitting that these axioms are coherent. It is also at the origin of the theory of the recursive Fonctions.

Generally regarded as Austrian, it was born with Brno in Austria-Hungary, naturalized Tchécoslovaque at 12 years, then Autrichien at 23 years. When Hitler orders the annexation of Austria, Gödel becomes German (it is then 32 years old). It leaves to the the United States during the Second world war, and it obtains the Austro-American dual nationality at 42 years.

It published its most important results in 1931 at the 25 years age, whereas it still worked for the Université of Vienna (Austria).

Biography

Childhood

Wire of Rudolf Gödel, leader of small a Undertaken Textile, and of Marianne Gödel (born Handschuh). Within this German-speaking family, small Kurt is called “Der Herr Warum” (Mr. Why). He attends the secondary elementary school then with Brno, which he finishes with the honors in 1923. Although Kurt initially excelled in languages, it becomes little of time later an enthusiastic amateur of history and mathematics. This passion for mathematics became new extensive in 1920 when his/her Rudolf older brother (born in 1902) left for Vienna to follow a medical course . Teenager, Kurt studies already work of Gabelsberger, the theory of Goethe on Isaac Newton, and the writings of Kant.

It is still at the University of Vienna that it meets that which will become (tardily) his wife, Adele Nimbursky (born Porkert). It publishes its first articles on logic and attends a conference of David Hilbert with Bologna on the complétude and the coherence of the mathematical systems. In 1929, Gödel becomes Austrian citizen before obtaining this same year its Doctorat, under the aegis of Hans Hahn. In its thesis, it establishes the complétude calculation of the first order predicates, result known under the name of Théorème of complétude of Gödel.

Studies Vienneses

At the 18 years age, Kurt joined his Rudolf brother at the University of Vienna. He at this time already acquired a university level in mathematics and philosophy. Although initially registered to study the Physical theoretical, it follows also a teaching in mathematics and philosophy. It is at that time that it adheres to mathematical realism. It reads Metaphysische Anfangsgründe der Naturwissenschaft of Kant, and joined the Cercle of Vienna where officiate Moritz Schlick, Hans Hahn, and Rudolf Carnap. Kurt studies the theory of the numbers thereafter, but turns quickly to mathematical logic after a seminar given by Moritz Schlick on the introduction to the philosophy of mathematics, of Bertrand Russell.

Work in Vienna

Gödel obtains its doctorate in philosophy in 1930. It proves in 1930 the complétude traditional logic first order, i.e. any valid formula is demonstrable, result which was published by the Academy of Science of Vienna. In 1931, it publishes its famous Théorème of incomplétude in Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme . It proves in this article that for any enough powerful axiomatic system to describe the natural numbers, one can affirm that:

1. It cannot be at the same time coherent and complete (what is the theorem known under the name of Théorème of incomplétude.)

2. If the system is coherent, then the consistency of axiom cannot be proven within the system.

These theorems reflect fine at centuries of attempts to propose a final set of axioms to locate the whole of mathematics on an axiomatic basis; with the manner of the Principia Mathematica and formalism of Hilbert. They also imply that there are mathematical questions which are valid, but which is not demonstrable.

The principle of the theorem of incomplétude is simple. Gödel primarily built a formula which states that it is not demonstrable in a given formal system. If this formula is demonstrable, then it is not demonstrable, from where contradiction. Thus this formula is not demonstrable, therefore valid. There thus exists a valid formula, nondemonstrable.

To specify these facts, Gödel needed to solve many engineering problems, like the coding of the demonstrations and the concept even of demonstrability within the integers. It, also needed a process to describe a formula which states its clean not demonstrability: the diagonal process . These details on the form explain why its publication of 1931 is also long and difficult to read and why its contemporaries except notable for John von Neumann and Alfred Tarski did not include/understand its result.

Gödel obtained its diploma at the University of Vienna in 1932, and became there Privatdozent in (lecturer) in 1933.

However, after the assassination on June 22nd 1936 of Moritz Schlick (of which the seminar had given birth to its interest for logic) by Hans Nelböck, a young student alienated, Gödel was particularly affected and crossed its first depression.

Travel to the United States

This year 1933 was also the occasion for Gödel to visit the United States, where it met Albert Einstein with which it bound a solid friendship. Later, it developed the idea Calculabilité, studied the recursive functions, so that it gave a conference on the general recursive functions and the concept of truth. This work was developed by using the construction of the numbers of Gödel.

In 1934, it gave conference series to the Institute for Advanced Study of Princeton entitled “Of the indecidability of the postulates of the formal mathematical systems”. Stephen Kleene and J. Barkley Rosser took in notes these conferences, published in the complete works of Gödel .

Gödel turned over to Princeton later the same year. The voyages and its work had exhausted it, so that the essence of the following year had to be devoted to the treatment of a new depression. It returned to teaching in 1937, period during which it worked on the proof of relative coherence and that of independence of the Hypothèse of continuous the. It failed on the independence (which will be shown by Paul Cohen only in 1963), but it succeeds in establishing that this assumption cannot be refuted starting from the axioms of the set theory. He married Adele the September 20th 1939 at the University of Notre-Dame.

Work with Princeton

After the Anschluss of 1938, Austria fell into the bosom from the Nazi Germany. The latter having abolished the title of Privatdocent , Gödel had to be concerned with an incorporation in the German armed . In January 1940, his wife and left to him Europe by the rail of the Transsibérien, going to the United States. After their arrival with San Francisco the March 4th 1940, Kurt and Adele settled in Princeton, where it reinstated the institute of the high studies of Princeton. To the institute, Gödel turned more still to philosophy and physics. He studied work of Gottfried Leibniz and, with a less degree, those of Kant and Edmund Husserl.

He continued his work of logician, and off published in 1940 The Consistency the Axiom off Choice and off the Generalized Continuum-Hypothesis with the Axioms off Set Theory . He introduces into this work concept of the constructible, model universe of the set theory in which the only existing units are those which can be built starting from more elementary units. Gödel proved that as well the axioms of choice and the assumption generalized of continuous are true in a constructible universe, and must thus be coherent. It had the intuition of the problems Np-complete S.

At the end of the years 1940, it showed the existence of a paradoxical solution to the equations of the general Theory of relativity of Einstein. The “revolving universes” would have made possible the voyage in time, and pushed Einstein to doubt its own theory (see Univers of Gödel). Today, this type of solution is regarded as a mathematical curiosity without much physical interest, but whose great merit is to have stimulated the search for other exact solutions to the equations of Einstein.

Become permanent member of the Institute of the studies advanced in 1946, it was naturalized American citizen in 1948. It obtained a post of professor at the Institute in 1953, refused the title of Professor emeritus in 1975 and was highly skilled in 1976.

In March 1951, Gödel accepted (at the same time as the physicist Julian Schwinger) the first Einstein price, then was named doctor honoris causa in several universities (Yale, Harvard, etc), and accepted the “Medal National off Science”, in 1974.

70 years old, Gödel, which was deeply believing, made circulate among his/her friends a development based on the ontological proof of the existence of God, inspired by the argument of Anselme of Canterbury and considerations of Leibniz. This development is now known under the name of ontological Preuve of Gödel .

Death and distinctions

Gödel was, throughout its life, a timid man and in withdrawal. Approaching the Dead, it felt increasingly concerned by its health, being convainquant of the existence of a plot aiming to the to poison. It then ceased feeding, falling gradually into the Cachexie. He died the January 14th 1978, in Princeton, state of the New Jersey, the United States.

The company Kurt Gödel, founded in 1987, was baptized in its honor. It is an international organization for the promotion of research in the fields of logic, philosophy, and the Histoire of mathematics.

A Prix Gödel which rewards best work in theoretical data processing was founded in its honor in 1992.

Anecdote

When Gödel arrived at the United States, this one had to undergo an examination for its naturalization. During its preparation, Gödel examined the American constitution and found inconsistencies logical in the latter which allowed, in all legality, to transform it political regime of the country in dictatorial mode. It announced its discovery to his friend Oskar Morgenstern who advised to him not to tackle the subject during his discussion with the officer of immigration.

Work

• Collected Works , Oxford University Close, 5 volumes published of 1986 to 2003 pennies direction of S. Feferman, J.W. Dawson, S.C. Kleene, G.H. Moore, R.M. Solovay and J. van Heijenoort.
• vol. I: Publications 1929-1936.
• vol. II: Publications 1938-1974.
• vol. III: Unpublished Essays and Readings
• vol. IV: Correspondence AG
• vol. V: Correspondence HZ

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