In logic, it refers to a theoretical system or proposition that can simultaneously deduce or prove two contradictory propositions. The definition of paradox can be expressed as follows: based on a proposition that is recognized as true, set it as B, and after correct logical reasoning, it is concluded that the proposition that contradicts the premise is not B; On the other hand, under the premise of non-B, B can also be deduced. Then proposition b is a paradox. Of course, non-B is also a paradox. We can judge or prove the truth of a proposition according to some formulated or agreed axiomatic rules, but when we judge or prove the truth of some propositions according to the formulated or agreed axiomatic rules, sometimes there will be insoluble paradoxes. What does this situation mean?

Nature as a whole contains diversity, but we ignore these situations and pay special attention to the special situations that belong to our interests. When a special situation encounters other opposite situations or general situations with universality, it will inevitably produce some contradictory conclusions. It is not the paradox of mathematics that has a great crisis impact on the basis of mathematics, but the great impact on logic and understanding.

Infinite set itself is a vague concept. Finite can be called a set, and infinite can't be called a set. Set refers to a certain range, and infinity refers to an infinite range, otherwise it should not be called infinity but finite. Infinity should not be an arbitrary choice or scope of application. If a quantity exceeds the level that human beings can reach or understand, it will enter the infinite range. So far, human beings have not fully and clearly known how big the radius we can identify is, so it is impossible to accurately and precisely define the boundary between infinity and finiteness.

The concept of set itself is an unrestricted concept. A total set can be arbitrarily divided into several sets, all of which are sets. To be exact, we don't know whether it is a set in that sense.

There are paradoxes in subsets, or with other sets, and there are also paradoxes between parent sets and child sets, because each specific subset has its own rules, and these rules are only valid within its own scope. Beyond the scope, it will be invalid, which is always inevitable or cancelled. Unless the classification of class set level is cancelled, it does not conform to the attitude towards specific things and cannot meet the practical application requirements. In addition, the original meaning and extended meaning of a set are often confused, and sometimes confused with the meaning of an element. A set is equivalent to a low-level element. When it rises, it is a set, and when it rises again, it is an element, which is cumulative.

Russell paradox is valid when they are not related. When they are interrelated, that is, they have become a class or a whole, then the two measurement standards or regulations are not allowed or can not be implemented in a class or a whole. Self-denial is the same as saying nothing, or it is equal to saying nothing.

Godel's completeness theorem and incompleteness theorem of first-order logic are paradoxes themselves, which expose the problems brought by logic. Godel's incompleteness theorem is a paradox caused by lack of judgment, taking the dominant aspect of decision-making as the standard, or too many standards. The so-called standards are also regulations. After failure, new rules can be stipulated according to actual needs. Anyway, the original rules are also stipulated. Why can't we redefine the rules to meet the needs of practical application after the paradox occurs? Obviously it is your own rule, but you create new rules to break the original rules. If you work like this, there will always be work to do, and there will always be endless work.

Classes are artificially distinguished, but classes are artificially created according to needs. If they are classified, the class is different. Generally speaking, there is nothing similar or different. Because of different classes, the figures are different, and some differences are opposite, which is normal and inevitable. However, if people want to switch between classes and numbers, they have to make new rules.

Prove that it is only in accordance with the pre-set and ideological regulations, and it will inevitably conform to the regulations. We just operate and execute according to the regulations. What is the function or significance of proof? Class paradox cannot be solved by proof.

Paradox is a part of a broad and strictly defined branch of mathematics, which is famous for "interesting mathematics". This means that it has a strong game color. However, don't think that all great mathematicians despise the problem of "mathematics is interesting". Euler laid the foundation of topology by analyzing the mystery of crossing the bridge. Leibniz also wrote about the pleasure of analyzing problems when he played the stick-inserting game (a game in which small pieces of wood were inserted into small squares) alone. Hilbert proved many important theorems in cutting geometry. Von Newman laid the foundation of game theory. The most popular computer game "Life" was invented by the famous British mathematician Conway. Einstein also collected a whole shelf of books about math games and intellectual games.

Paradox comes from the Greek word "para+dokein", which means "think more". The meaning of this word is rich, including all mathematical conclusions that contradict human intuition and daily experience, and those conclusions will surprise us. Paradox is a contradictory proposition. That is, if this proposition is admitted, it can be inferred that its negative proposition is established; On the other hand, if we admit the negative proposition of this proposition, we can deduce that this proposition is true. If the admission is true, after a series of correct reasoning, the conclusion is false; If you admit that it is false, after a series of correct reasoning, it is true. There are many famous paradoxes in ancient and modern China and abroad, which have impacted the foundation of logic and mathematics, stimulated people's knowledge and precise thinking, and attracted the attention of many thinkers and enthusiasts throughout the ages. Solving paradoxes requires creative thinking, and the solution of paradoxes can often bring people new ideas.

The earliest paradox is considered as the "liar paradox" in ancient Greece.

Principles for editing this paragraph

At the same time, it is assumed that two or more premises that cannot be established at the same time are the same characteristics of all paradox problems.

Generally speaking, because paradox is a formal contradiction, that is, the product of some special ideological provisions, it can not directly reflect the dialectical nature of things; Furthermore, we can't describe them as "special objective truths", but only as "distorted truths".

Therefore, paradox is essentially a concentrated expression of the contradiction between the dialectical nature of objective reality and the metaphysical and formal logical methods of subjective thinking. Specifically, as a part or side of the objective world, the research object of cognition or theory (mathematical theory and semantic theory) is often dialectical, that is, the unity of opposites; However, due to the limitation of metaphysics or formal logicalization in subjective thinking methods, the dialectical nature of objective objects is often distorted in the process of understanding: the link of unity of opposites is absolutely separated and exaggerated one-sidedly, so that it reaches an absolute and rigid level, thus dialectical unity becomes absolute opposition; And if they are mechanically reconnected, the direct conflict between opposing links is inevitable, which is paradox.

form

Paradox has three main forms.

1. An assertion seems to be definitely wrong, but it is actually right (paradox).

2. An assertion seems to be definitely right, but it is actually wrong (specious theory).

3. A series of reasoning seems unbreakable, but it leads to logical contradictions.

type

Paradoxes mainly include logic paradox, probability paradox, geometry paradox, statistics paradox and time paradox.

Russell's paradox shocked the whole field of mathematics with its simplicity and clarity, which led to the third mathematical crisis. However, Russell paradox is not the first paradox. Needless to say, not long before Russell, Cantor and Blary had discovered the contradiction in set theory in their forties. After the publication of Russell's paradox, a series of logical paradoxes appeared. These paradoxes remind me of the ancient liar paradox. That is, "I'm lying" and "this sentence is a lie". The combination of these paradoxes has caused great problems, prompting everyone to care about how to solve these paradoxes.

The first published paradox is Blary's Forty Paradox, which means that ordinal numbers form an ordered set according to their natural order. By definition, this well-ordered set also has an ordinal ω, which should belong to this well-ordered set by definition. However, according to the definition of ordinal number, the ordinal number of any segment in ordinal number sequence is greater than any ordinal number in that segment, so ω should be greater than any ordinal number, so it does not belong to ω. This was put forward by Blary Forti in an article read at the Balomo Mathematics Conference on March 28th, 1997. This is the first published modern paradox, which aroused the interest of the mathematical community and led to heated discussions for many years. There are dozens of articles discussing paradox, which greatly promotes the re-examination of the basis of set theory.

Blary Foday himself thinks that this contradiction proves that the natural order of this ordinal number is only a partial order, which contradicts the result ordinal number set proved by Cantor a few months ago. Later, Blary Foday didn't do this work either.

In his Principles of Mathematics, Russell thinks that although the ordinal set is fully ordered, it is not well ordered, but this statement is unreliable because the first paragraph of any given ordinal number is well ordered. French logician Jourdain found a way out. He distinguished between compatible sets and incompatible sets. This distinction has actually been used privately by Cantor for many years. Soon after, Russell questioned the existence of ordinal set in an article in 1905, and Zemelo also had the same idea, and later many people in this field held the same idea.

Paradox of classical mathematics

There are many famous paradoxes in ancient and modern China and abroad, which have impacted the foundation of logic and mathematics, stimulated people's knowledge and precise thinking, and attracted the attention of many thinkers and enthusiasts throughout the ages. Solving paradoxes requires creative thinking, and the solution of paradoxes can often bring people new ideas.

In this paper, paradox is roughly divided into six types, which are divided into three parts: upper, middle and lower.

The first part: the paradox caused by the concept of self-reference and the paradox brought by the introduction of infinity.

(A) the paradox caused by self-reference

In the following example, there is a problem of concept self-reference or autocorrelation: if we start from a positive proposition, we will get its negative proposition; If we start with a negative proposition, we will get its positive proposition.

1- 1 liar paradox

In the 6th century BC, the philosopher epimenides, a Crete, said, "All Cretes lie, and so does one of their poets." This is the origin of this famous paradox.

It is mentioned in the Bible: "A local prophet of Park Yung-soo said,' The Celts often lie, but they are evil beasts, greedy and lazy'" (Titus 1). It can be seen that this paradox is famous, but Paul is not interested in its logical solution.

People will ask: Is Epiminides lying? The simplest form of this paradox is:

1-2 "I'm lying"

If he is lying, then "I am lying" is a lie, so he is telling the truth; But if this is true, he is lying again. Contradictions are inevitable. A copy of it:

1-3 "This sentence is incorrect"

A standard form of this paradox is: if event A occurs, non-A is deduced; if non-A occurs, non-A is deduced, which is a self-contradictory infinite logic cycle. One-sided body in topology is the expression of image.

The philosopher Russell once seriously thought about this paradox and tried to find a solution. He said in the seventh chapter "Mathematical Principles" of "The Development of My Philosophy": "Since Aristotle, logicians of any school seem to be able to deduce some contradictions from their recognized premises. This shows that there is a problem, but it cannot point out the way to correct it. In the spring of 1903, one of the contradictory discoveries interrupted the logical honeymoon I was enjoying. "

He said: The liar paradox simply sums up the contradiction he found: "The liar said,' Everything I said is false'. In fact, this is what he said, but this sentence refers to all he said. Only by including this sentence in that crowd will there be a paradox. " (same as above)

Russell tried to solve the problem through hierarchical propositions: "The first-level propositions can be said to be those that do not involve the whole proposition; Second-level propositions are those that involve the whole first-level proposition; The rest is like this, even infinite. " But this method has not achieved results. "During the whole period of 1903 and 1904, I almost devoted myself to this matter, but it was completely unsuccessful." (same as above)

Mathematical principles try to deduce the whole pure mathematics on the premise of pure logic, explain concepts in logical terms, and avoid the ambiguity of natural language. But in the preface of this book, he called it "publishing a book that contains so many unresolved disputes." It can be seen that it is not easy to completely solve this paradox from the logic of mathematical basis.

Then he pointed out that in all logical paradoxes, there is a kind of "reflexive self-reference", that is, "it contains something about that whole, and this kind of thing is a part of the whole." This view is easy to understand. If this paradox is said by someone other than Park Jung-soo, it will be automatically eliminated. But in set theory, the problem is not so simple.

1-4 barber paradox

In Saville village, the barber put up a sign: "I only cut the hair of those people in the village who don't cut their own hair." Someone asked him, "Do you cut your hair?" The barber was speechless at once.

This is a paradox: a barber who doesn't cut his hair belongs to the kind of person on the signboard. As promised, he should give himself a haircut. On the other hand, if the barber cuts his own hair, according to the brand, he only cuts the hair of people in the village who don't cut their own hair, and he can't cut it himself.

So no matter how the barber answers, he can't rule out the internal contradictions. This paradox was put forward by Russell in 1902, so it is also called "Russell paradox". This is a popular and story-telling expression of the paradox of set theory. Obviously, there is also an unavoidable problem of "self-reference".

1-5 set theory paradox

"R is the set of all sets that do not contain themselves."

People will also ask: "Does R include R itself?" If not, according to the definition of R, R should belong to R. If R contains itself, R does not belong to R..

Kurt G?del (Czech Republic, 193 1) put forward an "incomplete theorem" after Russell's paradox of set theory found that there was a problem with the mathematical foundation, which broke the ideal of mathematicians at the end of 19 that "all mathematical systems can be deduced by logic". This theorem points out that any postulate system is incomplete, and there must be propositions that can neither be affirmed nor denied. For example, the negation of the "axiom of parallel lines" in Euclidean geometry has produced several non-Euclidean geometries; Russell's paradox also shows that the axiomatic system of set theory is incomplete.

1-6 bibliography paradox

A library compiled a dictionary of titles, which listed all the books in the library without their own titles. So will it list its own title?

This paradox is basically consistent with Barber's paradox.

1-7 Socrates paradox

Socrates (470-399 BC), an Athenian, is known as "Confucius in the West" and a great philosopher in ancient Greece. He was once opposed to the famous sophists Prut Golas and Gogis. He established a "definition" to deal with the confusing rhetoric of sophists, thus finding out hundreds of miscellaneous theories. But his moral concept was not accepted by the Greeks, and he was regarded as the representative of sophistry when he was seventy years old. Twelve years after expelling Prut Goras and burning books, Socrates was also executed, but his theory was inherited by Plato and Aristotle.

Socrates famously said, "I only know one thing, and that is nothing."

This is a paradox, and we can't infer from this sentence whether Socrates doesn't know the matter itself. There are similar examples in ancient China:

1-7 "Words are full of contradictions"

This is what Zhuangzi said in Zhuangzi's Theory of Everything. Later Mohism retorted: If "everything is against the truth", isn't Zhuangzi's statement against the truth? We often say:

1-7 "There is no absolute truth in the world"

We don't know whether this sentence itself is "absolute truth".

1-8 "absurd truth"

Some dictionaries define paradox as "absurd truth", and this contradiction modification is also a kind of "compressed paradox". Paradox comes from the Greek word "para+dokein", which means "think more".

All these examples show that logically, they can't get rid of the vicious circle brought by the concept of self-reference. Is there a further solution? We will continue our discussion in the last part of the next section.

(B) the introduction of infinite paradox

There is a saying in "Mo Zi Jing Shuo Xia": "There is poverty in the south, which can be exhausted; Endless. " If infinite is introduced into finite, it may cause paradox.