Difference between revisions of "Kurt Gödel"

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Kurt Gödel (1906-1978) was an Austrian mathematician who proved the greatest theorem of the 20th century: "Gödel's Incompleteness Theorems." This ended a century of attempts to place all of mathematics on an axiomatic basis.

Gödel published his remarkable proof in 1931. He showed that in any axiomatic mathematical system there are always propositions that cannot be proved or disproved using the axioms of the system. He additionally showed that it is impossible to prove the consistency of the axioms. This was the famous incompleteness theorem: any axiomatic system powerful enough to describe arithmetic on natural numbers cannot be both consistent and complete. Moreover, the consistency of the axioms cannot be proven within the system.

Godel's work abruptly ended a half-century of attempts, beginning with the work of Frege and culminating in Principia Mathematica and Hilbert's formalism, to find a set of axioms sufficient for all mathematics. The incompleteness theorems also imply that not all mathematical questions are computable.

This demonstrated the folly of the work of several liberal mathematical leaders. The self-described atheist Bertrand Russell had already published, in Principia Mathematica (1910-13), a massive attempt to do what Gödel later proved was impossible. Gödel's proof also disproved the entire "formalism" approach of David Hilbert.

Gödel's proof was a landmark for mathematics, demonstrated that it can never be a finished project as many mathematicians had believed. No one, not even the most powerful computer imaginable, can answer all mathematical questions.