Kurt Godel

Kurt Godel

Austrian logician and mathematician
Date of Birth: 28.04.1906
Country: Austria

Content:
  1. Biography of Kurt Gödel
  2. Education and Career
  3. Gödel's Dissertation and Contributions
  4. Contributions to the Theory of Sets
  5. Death

Biography of Kurt Gödel

Kurt Gödel was an Austrian logician and mathematician, known for his fundamental discovery that showed the limitations of the axiomatic method. He was born on April 28, 1906, in Brno.

Kurt Godel

Education and Career

In 1924, Gödel enrolled at the University of Vienna, where he pursued his studies in mathematics. In 1930, he successfully defended his doctoral dissertation in mathematics. From 1933 to 1938, Gödel served as a private lecturer at the University of Vienna.

In 1940, due to the political situation in Europe, Gödel emigrated to the United States. He continued his academic career as a professor at the Institute for Advanced Study in Princeton, New Jersey, from 1953 until his death.

Gödel's Dissertation and Contributions

Gödel's dissertation focused on the problem of completeness. Completeness of a system of axioms, which serves as the foundation of a particular area of mathematics, means the adequacy of that axiomatic system to the concepts of the mathematical area it defines. In other words, it means the ability to prove the truth or falsity of any meaningful statement containing the concepts of the considered area of mathematics.

In the 1930s, some results regarding the completeness of various axiomatic systems were achieved. David Hilbert constructed an artificial system that encompassed part of arithmetic and proved its completeness and consistency. Gödel, in his dissertation, proved the completeness of first-order predicate calculus, giving hope to mathematicians that they would be able to prove the consistency and completeness of all of mathematics.

However, in 1931, Gödel proved his incompleteness theorem, which dealt a devastating blow to these hopes. According to this theorem, any procedure for proving true statements in elementary number theory is doomed to be incomplete. Elementary number theory is a branch of mathematics dealing with addition and multiplication of integers. Gödel showed that, in any meaningful and practically applicable proof system, some truths even in such a modest area of mathematics will remain unprovable.

As a consequence, he concluded that the internal consistency of any mathematical theory can only be proved by appealing to another theory that uses stronger assumptions, and therefore, is less reliable. The methods employed by Gödel in proving the incompleteness theorem later played an important role in the theory of computation.

Contributions to the Theory of Sets

Gödel made significant contributions to the theory of sets. Two principles, the axiom of choice and the continuum hypothesis, eluded proof for decades, but their logical consequences remained attractive. Gödel proved in 1938 that adding these principles to the usual axioms of set theory does not lead to contradiction. His reasoning was valuable not only for the results it allowed to obtain but also for improving the understanding of the internal mechanisms of set theory itself.

Death

Kurt Gödel passed away in Princeton on January 14, 1978, leaving behind a lasting legacy in the fields of logic, mathematics, and theoretical computer science.

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