Godel is a leader in understanding the narrative characteristics of mathematics. This way of looking at things opens a new branch of mathematics-abstract mathematics. The commonly used mathematical analysis methods are related to the imitation-germination stage of abstract mathematics, which forms the essence of formal system-mathematics itself is assumed to be the original sample of abstract mathematics. In this way, mathematics is like a snake that has eaten itself. It turns its head to catch itself. Godel shows that the strange conclusion comes from the focusing process when looking at mathematics itself through the mathematical lens. All the symbols used to write legendary works happen to be Arabic numerals from 0 to 9 that we usually use. Therefore, Martians will discuss a famous discovery in textbooks. They will find that we are related to Euclid on the earth, and at the same time we will say that "there are many prime numbers in their works", and what they write will be like this: "445945085 looks like 46 digits to us. For Martians, it is not a number at all, but a statement. Indeed, for them, the prime numbers they wrote represent 34 letters, 6 words and a few lines, just like I used English letters with you. Now let's imagine discussing the universal properties of all mathematical theorems. If we look up Martian textbooks, all theorems we see are just pure numbers. So we may create a complicated theorem to tell us which numbers can appear in Martian textbooks, and those numbers will never appear there. Of course, we don't want to talk about numbers, but prefer to talk about symbol chains that look like numbers. Moreover, perhaps it is not easy for us to forget the significance of these symbol chains to Martians, but just regard them as ancient numbers.

Through this simple perspective of transposition, Godel found a more profound force method. Godel's method is to imagine what can be called "numbers created by Martians" (those numbers are actually theorems in Martian textbooks), and he tries to ask the question: "Was 8030974 created by Martians?" The question is, will a narrative like "8030974" appear in Martian textbooks? Godel carefully pondered the surreal number composition, and soon he found that this special number created by Martians was not completely different from the familiar concepts such as "prime number" or "odd number". In this way, the number theory within the scope of the earth can deal with such problems as "which numbers were created by Martians and which numbers were not created by Martians" or "whether there are infinite non-Martians to create numbers". It is likely that advanced mathematics textbooks (on earth) have included all the digital sources created by Martians. In this way, Godel designed an amazing sentence with one of the most keen insights in the history of mathematics: "X is not a number created by Martians." X in this sentence is the number when the sentence "X is not a number created by Martians" is translated into the mathematical concept of Martians. Think about this sentence carefully until you understand it. Translating the sentence "X is not a number created by Martians" into the concept of Martians will be a huge number chain for us-a big number, but this string of Martian characters is exactly what we are looking for (X mentioned in this sentence itself). It's so tortuous, it's really, really tortuous! But twists and turns are Godel's specialty-twists and turns are in the spatial structure, twists and turns are in the reasons, and everything is tortuous. By turning ideal into a symbolic pattern, Godel found that the statement expressed by "formal system" can not only clarify himself, but also reject his theoretical source. The potential result of this mathematical entanglement is a great and unusual sorrow for Martians. Why are you sad? Because people on Mars-like Russell and Whitehead-have long hoped that their formal system can master all the true statements of mathematics. If Godel's statement is correct, it will not be regarded as a theorem in their textbooks, nor will it appear in their textbooks again-because Godel's statement has shown that it is impossible in itself! If it does appear in their textbooks, what is the wrong explanation for it? Who, even Martians, would want a math textbook that advocates both mistakes and correctness? As a result of all this, the formalistic goal that has been maintained is nothing more than an illusion. All formal systems show that they are incomplete because they can show that they cannot be proved. Moreover, it is said that Godel's "mathematical incompleteness" in 193 1 also illustrates the above viewpoint. In fact, it is not that mathematics itself is incomplete, but that any formal system that tries to grasp all the facts of mathematics with a limited set of axioms and rules is incomplete. For you, this conclusion may not shock you, but for mathematicians in the 1930s, it ended their whole world outlook, and mathematics was unrecognizable from then on. The article written by Godel in 193 1 also had other influences: it invented the theory of cyclic function and became one of the important basic theories of computer theory today. Indeed, at the core of Godel's article, a complex approximate computer program for creating "Martian-created" numbers was written in the form of a programming language very similar to Lisp, which was developed nearly 30 years later. Godel is as strange as his theory. 1939, he and his wife Etienne, a professional dancer, fled Nazi Germany and went to Princeton. There, he worked with Einstein in the Institute of Advanced Studies. In his later years, Godel became a paranoid patient about bacterial infection. He washed the dishes over and over again, wearing a ski mask and running around with his eyes exposed. For a time, he became a notorious figure. At the age of 72, he died in Princeton Hospital because he refused to eat. Just as the power of the formal system is doomed to be incomplete, so is life. Just as the complexity of the formal system is doomed to perish, everyone has his own unique way of life.