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202 1-0 1-22 the achievement of modern mathematics-godel's incompleteness theorem

Godel's Incomplete Theorem, also known as Godel's Incomplete Theorem, is two theorems proved and published by Kurt G?del in 193 1. Godel's incompleteness theorem destroyed the philosophical attempt of Hilbert Plan. David hilbert suggested that the compatibility of a complex system like real analysis can be proved by a simpler system. In the end, the compatibility of all mathematics can be attributed to the compatibility of basic arithmetic. But Godel's second theorem proves that the compatibility of basic arithmetic cannot be proved by itself, so it cannot be used to prove the compatibility of systems stronger than it. It has greatly promoted the development of mathematics and multidisciplinary theory.

Unprecedented discussion on basics is one of the trends of pure mathematics in the 20th century, and the important motivation of this discussion is compatibility. Godel's incompleteness theorem is a very profound ideological achievement put forward by studying the compatibility theorem. 193 1 year, Austrian mathematician godel published the article "Principles of Mathematics", and put forward the proposition that forms cannot be rebelled in related systems, in which godel's incompleteness theorem was put forward.

If any formal system containing natural number arithmetic is compatible, there must be an irresistible proposition, that is, the existence and negation of a proposition A cannot be proved in this system.

These axiomatic compatible (non-contradictory) propositions are contained in true propositions that cannot be proved by axioms. In other words, an axiomatic system containing natural number arithmetic is compatible, so this compatibility is unprovable in this system.

Godel 1906 was born in Brno, Czech Republic (the former Austria-Hungary), graduated from Vienna University, 1940 immigrated to the United States, and worked at the Institute for Advanced Studies (IAS) in Princeton until 1976 retired. 1978, Godel died in Princeton, New Jersey. Godel has been sick since childhood, suffering from obsessive-compulsive disorder (hypochondria) since childhood. He also suffers from depression. Later, he died of hunger strike in Princeton Hospital because he thought the food was poisonous.

Most of Godel's manuscripts were written in shorthand by Gabes Berg. Motto: "Some facts are recognized as true, but they are not necessarily provable."