Let's look at a sentence first: "I'm lying." Is this sentence a lie or a truth? This famous liar paradox has actually touched Gotham's incompleteness theorem. Let's look at another sentence B: "This sentence is false". Is sentence B true or false? If you decide that sentence B is true and sentence B is false. If you decide that sentence B is false, sentence B is true. So, you can either accept that sentence B is both true and false. Accept it or not: the true value of sentence B cannot be judged. Gotham pointed out that in any formal system with strong expressive force, we can construct a proposition T similar to sentence B, which makes; That is, t can be proved if and only if t cannot be proved; That is, the meaning of T is "the proposition of T cannot be proved". By constructing T, Gotham obtained his "first incompleteness theorem": any formal system with strong expressive force cannot be consistent and complete at the same time. What kind of system is complete? If the truth values of all expressible propositions in a system can be determined, either true or false, then we say that the system is complete. For example, in the arithmetic system, the proposition1>; 2 is false, proposition 3 >; 2 is true and the proposition is false. If all propositions formed by second-order logic, numbers and their comparison symbols can be judged as true or false, then this arithmetic system is complete. What kind of system is consistent? A system that never allows "contradictions" is consistent. Paradoxically, for example in an arithmetic system, if 1 > is not allowed at the same time; 2 and 1