The popular explanation of Godel's incompleteness theorem is that in the natural number system, "self-consistency" and "completeness" cannot have both, so we can only give up one and save one, that is to say, we can't have both.

This theory has made epoch-making changes in the basic research of mathematics and is a very important milestone in the history of modern logic. This theorem, together with Talsky's theory of formal language truth, Turing machine and judgment, is regarded as the three philosophical achievements of modern logic science. Godel proved that any formal system, as long as it contains a simple elementary number theory description and is self-consistent, must contain the proposition that the methods allowed in some systems can neither prove truth nor falsify.

Influence of Godel's Incompleteness Theorem

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. "