There are questions about mathematical paradox.

Zeno Paradox is a series of philosophical paradoxes about the inseparability of motion put forward by Zhi Nuo, an ancient Greek mathematician. These paradoxes are known to later generations because they are recorded in Aristotle's book Physics. Zhi Nuo put forward these paradoxes in order to support his teacher parmenides's theory that "being" is fixed and a "being". The two most famous paradoxes are: "Achilles can't outrun the tortoise" and "the arrow doesn't move". These methods can now be explained by the concept of calculus (infinity).

Sets can be divided into two categories: the first category is characterized by: a set itself is an element in a set, such as the "set composed of all sets" that people often said at that time; The characteristic of the second set is that the set itself is not an element of the set, such as a set of points on a straight line. Obviously, a set must be one and only one of these two kinds of sets. Now let's assume that R is the set of all sets of the second kind. So, what kind of set is R?

Russell's paradox

If R is the first kind, R is its own element, but according to the definition, R only consists of the set of the second kind, so R is also the set of the second kind; If R is a set of the second kind, then according to the definition of R, R must be an element of R, so R is also a set of the first kind. In short, I am in a dilemma and can't give an answer. This is the famous "Russell paradox".

Examples of Russell Paradox

Russell paradox example

There is a story in the world literary masterpiece Don Quixote:

Sancho Panza, the servant of Don Quixote, ran to an island and became the king of the island. He made a strange law: everyone who arrives on this island must answer a question: "What are you doing here?" If the answer is right, let him go to the island to play, if the answer is wrong, hang him. For everyone who comes to the island, they will either have fun or be hanged. How many people dare to risk their lives to play on this island? One day, a bold man came. Ask him this question as usual, and the man's answer is: "I'm here to hang myself." Will Sancho Panza let him play on the island or hang him? If he should be allowed to play on the island, this is inconsistent with what he said about being hanged, that is, what he said about being hanged is wrong. Since he is wrong, he should be hanged. But what if Sancho Panza wanted to hang him? At this time, what he said "to be hanged" was true and correct. Since he answered correctly, he should not be hanged, but should be allowed to play on the island. The king of this island found that his laws could not be enforced, because they would be destroyed anyway. He thought and thought, and finally let the guard let him go and declared the law invalid. This is another paradox.

The paradox put forward by the famous mathematician Bertrand Russell (Russel, 1872- 1970) is similar:

There is a barber in a certain city. His advertisement reads: "My haircut skills are superb and the whole city is famous. I will shave all the people in this city who don't shave themselves. I will only shave these people. I would like to extend a warm welcome to everyone! " When people come to him to shave, they naturally don't shave themselves. One day, however, the barber saw in the mirror that his beard had grown. He instinctively grabbed the razor. Do you think he can shave himself? If he doesn't shave himself, then he belongs to the "person who doesn't shave himself" and he has to shave himself. What if he shaved himself? He belongs to the "person who shaves himself" and should not shave himself.

Barber paradox and Russell paradox are equivalent;

Because, if everyone is regarded as a set, then the elements of this set are defined as the objects that this person shaves. Then, the barber claimed that his element was all the collections in the village that did not belong to him, and all the collections in the village that did not belong to him. So does he belong to himself? Thus, Russell's paradox is obtained from Barber's paradox. The same is true of reverse transformation.

Liar paradox and liar cycle are paradoxes closely related to natural language expression, which involve semantic concepts such as truth and falsehood, definition, name and meaning, and are called "semantic paradoxes". There are many examples of semantic paradox, and the "K.Grelling)- L.Nelson Paradox" is very interesting, which is related to the application of adjectives:

Adjectives are divided into two categories, one is called "self-reference", that is, being true to oneself. For example, the adjective "polysyllabic" itself is polysyllabic, and "English" itself is English. It's all self-referential The other is called "what does it say", which means that it is not true to itself, and it is not true to itself. For example, the adjective "monosyllabic" means this, because this word is not monosyllabic; "English" also means this, because the word is Chinese, not English. The question is: Is the adjective "it refers to" what it refers to?

The result is that if it says what it says, it will come to the conclusion that what it says is not what it says, and vice versa. Leading to self-contradiction.

Paradox and Axiomatization of Set Theory

Another paradox involves set theory in mathematics, which is called "mathematical paradox" or "set theory paradox". Set theory was founded by German mathematician Cantor in 1970s and 1980s. It is based on an infinite view-"real infinity". The so-called "real infinity" means taking "infinity" as a complete conceptual entity. For example, in set theory, n = {n: n is a natural number} is used to represent the set of all natural numbers. It should be pointed out that in the thousands of years of mathematical development before this, another view of infinity dominated, that is, the concept of "infinite potential" advocated by the ancient Greek philosopher Aristotle. The so-called "potential infinity" is to regard "infinity" as a process that is constantly developing and can never be completed. For example, think of natural numbers as an infinite sequence 1, 2, 3, …, n, … that's it.

Set theory is a revolutionary change in mathematical concepts and methods. Because it is very convenient to explain old mathematical theories and develop new ones, it is gradually accepted by many mathematicians. However, shortly after Cantor founded the set theory, he himself discovered this problem, which is the "Cantor Paradox" of 1899, also known as the "Maximum Cardinal Paradox". At the same time, other set theory paradoxes have also been discovered, the most famous being "Russell Paradox" in 190 1:

Sets are divided into two categories, and any set that does not take itself as an element is called a normal set (for example, the set of natural numbers n itself is not a natural number, so n is a normal set. Any set with itself as an element is called an exception set. (For example, all abiotic sets f are not living things, so f is an abnormal set. ) Each set is either a normal set or an abnormal set. Let v be the set of all normal sets, that is, v = {x: x? X}, so is v a normal set?

If V is a normal set, how do you know V from the definition of normal set? V, because V is the set of all normal sets, so the normal set V∈V, but this shows that V is not a normal set, but an irregular set; On the other hand, if V is not a normal set, but an irregular set, then V∈V can be known from the definition of the irregular set, which shows that V is an element of the set V composed of all normal sets, so V should be a normal set.

Russell's paradox reveals a harsh fact: set theory implies logical contradictions. If mathematics is based on set theory, it will cause deep cracks in the foundation of the mathematics building, and may even overturn the whole building. A stone stirred up a thousand waves and broke out an argument about the basic problems of mathematics.

In this debate, the most radical intuitionistic school, represented by Dutch mathematician Brouwer, took a completely negative attitude towards set theory, thinking that the concept of "real infinity" is the root of the paradox of set theory. On the contrary, other mathematicians have embarked on the road of improvement, trying to make up for it and make appropriate amendments to set theory to avoid paradox. The representative achievement in this respect is axiomatic set theory, which has become an important branch of modern mathematics. Axiomatic set theory uses axiomatic method to describe sets and their operations, and modifies the "generalization principle" in Cantor's set theory. The generalization principle can be expressed as follows: all objects satisfying the property P can form a set S, that is, S = {x: P(x)}, where P(x) means "X has the property P". This proves that any property can determine a set, so the aforementioned f and v become a set, and the paradox arises at the historic moment.

In the ZF system of axiomatic set theory, the generalization principle is replaced by the following "separation principle": If C is a set, those elements in C that satisfy the property p form a set S = {x: x ∈ C, P(x)}, that is, any property can determine a subset of C on the premise that C is a set. As a result of axiomatization, only normal sets can become sets, abnormal sets can't, and F and V are not sets, which can avoid Russell paradox and other set theory paradoxes.

As far as axiomatic set theory can avoid the existing paradox of set theory and further develop mathematics on this basis, it is successful Unfortunately, people can't prove the compatibility of axiomatic set theory system, that is, they can't prove that logical contradictions can't be deduced in the system. In addition, some results in modern mathematics need to use "axiom of choice", but this will lead to some counterintuitive theories (such as "the theory of dividing the ball"). Therefore, it is necessary to further discuss the treatment of axiomatic set theory, especially the use of axiom of choice.

Some in-depth discussions on paradox

The discovery of Russell's paradox also promotes the in-depth thinking on the causes of paradox (including semantic paradox). During 1905- 1906, poincare put forward the conclusion that the root of paradox lies in the definition of indirect predicate in mathematics and logic. The so-called indirect definition refers to defining a concept (or object) with the help of a whole, and this concept (or object) itself belongs to this whole. This definition is cyclical (Russell called it "vicious circle") or "self-involvement". For example, the exception set "all abiotic sets F" is like this. Because F is defined as the whole of "all nonliving things", and F itself is a member of this whole. Investigating semantic paradox, we will also find similar traces of "circulation" or "self-involvement". For example, "liar cycle" refers to the mutual circulation of A and B, while the definitions of "self-reference" and "other-predicative" in Glenn-Nelson paradox involve the truth or falsehood of adjectives.

193 1 year, Talsky put forward the theory of "language hierarchy" in his article The Concept of Truth in Formal Language. Although this theory is mainly aimed at formal language, it is also of great significance to the study of semantic paradox in daily language. Talsky believes that everyday language is semantically closed: it contains both linguistic expressions and statements (such as "true" and "false") that state the semantic properties of these linguistic expressions. This is the root of semantic paradox. In order to establish a proper definition of "true sentence" which is correct in essence and form, it is necessary to deal with the language in layers: the sentence under discussion belongs to a certain level of language (called "object language"), while the sentence stating the semantic nature of the sentence belongs to a higher level of language (called "metalanguage"). The "liar paradox" is caused by asserting one's own truth and confusing the level of language.

1975 S.A. Kripke, a famous contemporary logician, put forward a new solution to paradox in the article Outline of Truth Theory. One of the core concepts is "rootedness": to judge a statement with a truth predicate ("true" or "false"), we must find the "root" of this statement-the corresponding statement without a truth predicate. For example, judging whether the sentence "water is colorless and transparent" is true or not depends on whether the sentence "water is colorless and transparent" is correct. The last sentence contains no truth predicate, so it can be judged right or wrong. So the previous sentence has a root. Only a sentence with a root can judge whether it is true or not, but a sentence without a root cannot. The "liar paradox" and "liar cycle" are rootless, which is the basic feature of paradox.

Recently, the study of paradox has been influenced by situational semantics. Linguistic logicians have noticed that many semantic paradoxes are not only related to semantics, but also closely related to pragmatic factors such as the context (including language users) when speaking. Take "liar paradox" as an example. When a person says "I'm lying", it means that he expresses the assertion that this sentence is true in a certain context. But the statement "I'm lying" is false, but it can't be stated in the same context and in another context. Therefore, the root of paradox lies not in "self-involvement", but in different contexts. As long as the context of each sentence is clear, many so-called "paradoxes" are no longer real paradoxes.