Reflections on the paradox of Set Theory

Document directory

• Russell Paradox
• Paradox solution
• Some philosophical difficulties
• Vitegenstan's new ideas
• Defects of new ideas

Russell Paradox

Speaking sets are the most important theory in modern mathematics and are the cornerstone of many other theories. we know that various things are classified in daily life. then, we determined what belongs to this category and which does not. there is a clear boundary. the set in mathematics is similar to classifying integers, decimals, irrational numbers, and other mathematical objects according to some rules. A set that meets certain rules. different sets may have their own unique properties. for example, we can regard all prime numbers as a set.

However, when talking about the classification of sets, we have to talk about the widely known Russell paradox. there are some classification standards that make it impossible to define a boundary for a set. It cannot be determined whether an element belongs to it or not. in the logic, only right and wrong are selected. if there is a third choice, the entire logic theory will be messy. the Russell paradox can be explained in a simple story. there was a barber in a village. He said he only gave a haircut to those who never cut their hair.

Now the problem arises. If we need to use the above rules to determine a set, then we divide the people in the village into those that belong to the Set (meeting the conditions for haircutting) and those that do not belong to the set. which category does the barber divide himself? It seems that it is wrong to randomly classify the class. This will confuse the mathematicians.

Paradox solution

Finally, I don't know what specific mathematical methods are used to solve the problem. the final solution seems to be like this. If such a paradox exists in a proposition, it indicates that this proposition is wrong. not logical. let's assume that we found that the question was wrong when we were doing a math problem, so we could not solve it.

 

Some philosophical difficulties

Some ideas proposed by vitegenstein in logical philosophy are actually a bit similar to those mentioned above. for thousands of years, philosophers have not been able to discuss the ultimate philosophical issues, such as what is the most fundamental component of the world, and what is the concept of horse. the root cause of the incident (that is, there is another reason behind any reason )... there are many strange issues. some people try to give some answers, but others may not agree.

For example, in ancient Greece, many people thought that the world was composed of some basic particles. but the problem is that each particle itself does not have some properties, but it is strange to combine it. it may involve both power and energy. However, force and energy seem to be virtual, and we cannot see them. Our subjective thinking is hard to imagine. in addition, Einstein's intelligent equation assumes that mass loss can be converted into energy.

What is the abstract concept of the horse? Plato believes that there is a world of ideas. in reality, the specific horse is simulated based on the horse in the concept world. some people think that the world where God is.

There seems to be a discussion about the first cause. I can't remember what I said. it seems that at the end of this article, the material itself has some characteristics that lead to a causal relationship. for example, the reason why the earth turns around the sun is gravity. where did gravity come from? Dumb: Some people may say that some properties of matter produce gravity. But where are those properties? As a result, a substance itself has this nature. however, many people may not be able to come up with such an answer. They can only say that there is a omnipotent god. then the antitheists will ask where God came from? There is no final answer at the end.

 

Vitegenstan's new ideas

The above-mentioned similar issues have not been resolved for thousands of years. in modern times, a buddy called vitegenstan made an innovation and tried to answer these questions from another distinctive point of view. gossip first.

He was born in 1889. He was born in the same year with him. Some legends were Hitler, and they were still classmates. In addition, he was also born in the same year, and he grew up like Hitler.

The preceding solutions are similar to those of Russell's paradox. he believes that the world can be divided into facts, facts can be divided into atomic facts, and atomic facts are composed of objects and things. an atomic fact corresponds to a proposition. true or false when the proposition is consistent with the reality. the true and false propositions cannot be judged. Just like the problems in the Russell paradox, we should not discuss them. we should keep silence on such issues. in fact, we cannot say that such a proposition is wrong, because it completely violates the logic, so we cannot use logic to discuss right and wrong. all his arguments can be summarized in this simple way, and he tries to divide all his propositions (including all philosophical propositions) into two categories. One can be said, it can be clearly stated in the logical scope and discussed clearly. another type of proposition cannot be discussed. It cannot be discussed using logical concepts to separate right and wrong.
As a result, there is no need to compete for many philosophical propositions, because they cannot be said and cannot be argued. we can only keep silence. when talking about "not to mention", many people think about some ideas in Zen. in the legend of Yi Tian tu long, Mingjiao does not have a bag Monk to say no, but zhangkou cannot say or say no. therefore, the Zen thought that many things do not need to be discussed and cannot be identified by logic. this is similar to what vitegenstein said.

 

Defects of new ideas

However, vitegenstan's philosophical point of view also has a problem. from his point of view, only propositions in natural science can be said. ethical Issues in daily life cannot be said. however, for our lives, such problems, that is, the unspeakable problems seem to be more important than the logic problems that can be said. many territorial disputes are naturally hard to be clarified by logic theory (not that class ). what's the final result? Naturally, it is reasonable to explain whether the fist is hard and the national strength is strong.

Vitegenstan's late stage also reflected on his early thoughts. he started his research on natural language and discussed some philosophical questions from some details of natural language. there is no complete system. In fact, books are just scattered essays. may have died before writing.