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Incompleteness Theorem

The Incompleteness Theorem is a pair of logical proofs that revolutionized mathematics. The first result was published by Kurt Gödel (1906-1978) in 1931 when he was 24 years old.

The First Incompleteness Theorem states that any contradiction-free rendition of number theory (a branch of mathematics dealing with the nature and behavior of numbers and number systems) contains propositions that cannot be proven either true or false on the basis of its own postulates. The Second Incompleteness Theorem states that if a theory of numbers is contradiction-free, then this fact cannot be proven with common reasoning methods.

Some mathematicians found Gödel's proofs disturbing when they were published. Today, serious students of mathematical logic find them fascinating. Some people have seized upon Gödel's results and attempted to apply them to nature in general, to social science, and even to theology. Many of these extensions of Gödel's results are inappropriate; a few are, by scientific standards, ridiculous.

Kurt Gödel was born in the Czech Republic and grew up in Austria (which was the Austro-Hungarian Empire in Gödel's early childhood). His primary language was German. Although he is most famous for his contribution to mathematical logic, he also did much work in set theory . He was a friend of Albert Einstein during the time they were both at the Institute for Advanced Study at Princeton University

This was last updated in September 2005

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