Mathematical incompleteness theorem refers to that in mathematics, some propositions cannot be proved or falsified, that is, their authenticity cannot be determined. This theorem was put forward by German mathematician Godel in 193 1, which is called Godel's incompleteness theorem.

Godel's incompleteness theorem is mainly divided into two parts. The first part is about a formal mathematical system. If this system is self-consistent (no contradiction), then it must contain some propositions that cannot be proved true or false. The second part is aimed at natural language, and any mathematical statement in natural language can't be proved or falsified, that is to say, natural language can't completely describe the authenticity of mathematics.

The proposition of this theorem has a profound influence on mathematics and philosophy, which proves that some problems in mathematics cannot be completely solved, and also reveals the limitations of formal mathematics. At the same time, this theorem has also become an important inspiration for the development of computer science and artificial intelligence, because it proves the essential difference between human thinking and machine intelligence.

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Godel's incompleteness theorem shattered the belief of mathematicians for two thousand years. He told us that truth and provability are two concepts. What can be proved must be true, but it is not necessarily true.

In a sense, the shadow of paradox will always be with us. No wonder the great mathematician William lamented: "God exists because mathematics is undoubtedly compatible;" The devil also exists, because we can't prove this compatibility. "

But the influence of Godel's incompleteness theorem goes far beyond the scope of mathematics. It not only revolutionized mathematics and logic, but also caused many challenging problems. It also involved philosophy, linguistics, computer science and even cosmology. August 2002 17.