paradox that can be derived logically contradictory conclusions, but the surface can justify the proposition or theoretical system. Paradox occurs because people tend to understand some concepts lack a deep understanding correctly. Paradox is extremely complex and profound causes of their in-depth research helps math, logic, semantics, and so theoretical development of the subject, it is of great significance. One of the most classic paradox, including Russell's paradox, the Liar paradox, Cantor paradox and so on.
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concept principle of mathematical paradox of the classic form of the type 1-1, 1-2 Paradox lies 1-6 division titles paradox paradox Paradox of Set Theory 1-5 1-7 1-8 Socratic Paradox sophistry? 5-1 the egg? should pay the tuition fees? 5-12 Vatican scholars of the noise floor is not the whole bag of millet, 10 Why is the sky dark? Postscript
Concept bèilùn (paradox, also known as inverse theory, anti-s) of logic or evidence that can derive two contradictory propositions propositions or theoretical system.
paradox can be defined this way: to be recognized by a true proposition as the premise, set B, the correct logical reasoning, the derive a prerequisite for interaction with conclusions for the non-contradictory propositions B; the contrary, the premise of non-B can also push B. Then the proposition B is a paradox. Of course, non-B is also a paradox. We can make or agreed in accordance with certain axioms or proof rules to determine the truth of a proposition, but we follow the axioms developed or agreed upon rules to determine or prove the truth of some proposition, sometimes it can not be resolved there occurred Paradox, this situation illustrates the problem? Is included in the overall natural diversity, and we set the expense of these situations, and special attention is that we are interested in a special case, when the special circumstances of the case or the other opposite the universality of the general situation there met When bound to produce some contrary conclusions. Not a mathematical paradox of mathematics have a big crisis, but a significant impact on the logic and understanding. Infinite set itself is a vague concept of requirements is called a limited set of infinite sets is not known. Collection is in a certain range that it refers to the range of infinite infinity, otherwise known as the infinite and said should not be limited. Should not be an unlimited arbitrary choice or scope of application, a number that can be achieved when more than human understanding of the extent or will enter the infinite scope. Until now, humans have not fully clear that we can recognize how big the radius, so the provision can not be accurately infinite and the finite precision the boundaries between them what is there. The concept of the collection itself is a concept not restricted, the total collection can be arbitrarily divided into several collections, all collections, we do not know what exactly is the premise in that sense the set of constraints. Subset of paradoxes, or between any other set of paradoxes,
timberland boots men, picture-collection still exists between the paradox, because in each there is a specific subset of the provisions of its own rules, only in their own the range is valid. Beyond the scope of the failure, which is always inevitable or canceled. Cancel the class unless the difference between the set level, then things are not in compliance attitudes towards specific, can not meet the requirements of practical application. Another collection of original meaning and extended meaning often mixed and sometimes confused with the elements of significance, the equivalent of a collection of elements in the low-level, when the rise is set to rise again when Shiyou equivalent elements are cumulative. Russell's paradox when they are not in contact with each other is effective, the time when they contact each other that they have become a class or a whole, then a class or a whole is not allowed or can not perform two or measure provisions, and did not say self-denial is a kind, or equal to not require the same. Gödel's completeness on the first-order logic theorem and Incompleteness Theorem is a paradox in itself, has exposed the logical cause of the problem. Gödel's Incompleteness Theorem is the lack of judges to determine the dominant aspects of the measure, or measure of the paradox caused by too much. Is a so-called standard requirements. After the failure can also be based on actual needs new rules again,
asics shoes sale, anyway, is the original provisions of the rules, why there can not occur again after the paradox of making the required rules, the purpose of practical applications to meet the needs? Clearly the provisions of their own, but they create their own new rules to undermine the provisions of the original, and if so to work, then never did live, live forever piling up. Class is distinguished from man-made, but the classes are arbitrary man-made necessary, if the classification, so the class is different. Does not exist on the whole similar and different, because different types, so the numbers are different, contrary to some different is normal and inevitable. However, the number of people between classes and want to transform, so they had to make new provisions. Proved only in accordance with the pre-set to operate and the provisions that are bound to comply with regulations, we just perform the required operation well, prove what effect or significance? Paradox not by class to prove to our best to resolve individually. Paradox of the field are broad definition of the strict branch of mathematics is an integral part of this branch of the This means that it is a game with a strong color. However, the refrain that the great mathematicians have looked down on Euler is through the mystery of the bridge-crossing analysis of the topology of the foundation laid. Leibniz also wrote that he inserted a stick in the game to play alone (a plug in a small wooden box in the section of the game) when analyzing the problem of fun. Hilbert proved that cutting a number of important theorems in geometry. Foundation of game theory von Neumann. The most popular computer game - Life is a famous British mathematician invented Conway. Einstein also had a whole shelf collection of math games and math on the mystery of the book. Paradox (paradox) from the Greek Rich meaning of this word, which includes people with all the daily experience of intuition and the mathematical contradictory conclusions, those conclusions will make us very surprised. Paradox is self-contradictory proposition. If it is recognized that this statement is true, you can set up launch of its negative proposition; the other hand, if the recognition of the negation of this proposition statement is true, but also set up to launch this proposition is true if it is admitted, after a series of correct reasoning, but may that it is false; admit that it is false if, after a series of correct reasoning, but come it is true. Many well-known paradox of ancient and modern, which shook the foundation of logic and mathematics, and aroused the people to knowledge and sophisticated thinking, through the ages attracted the attention of many thinkers and lovers. Solve the paradox of problems require creative thinking to solve the paradox often can bring new ideas. The first paradox is that the ancient Greek Generally speaking, the paradox is a form of contradiction, that is, the idea of certain provisions of the special product of things they can not be a direct reflection of the dialectical nature; then, we will not put them as objective truth, Therefore, the paradox is essentially the dialectic of objective reality and subjective metaphysical thinking and formal logic of the method of the concentrated expression of the contradiction. Specifically, as a part of the objective world or side, knowledge or theory (mathematics, semantics) of the object in nature is often dialectical, that is, the unity of all opposition links; However, due to the subjective way of thinking the metaphysical logic of the method or form of the restrictions, the objective nature of the object in the understanding of this dialectical process often been distorted: the link is absolute unity of opposites separated, and was one-sided exaggeration, that to achieve the absolute, the degree of rigidity to the dialectical unity of the opposition becomes absolute; and if then they are mechanically linked together again, against aspects of direct conflict is inevitable, and this is the paradox. There are three main forms of the form of paradox. 1. Looks like a thesis must be wrong, but in fact it is right (paradox). 2. Looks like a judge must be right, but in fact it was wrong (plausible theory). 3. Looks like a series of impeccable reasoning, but they lead to logical contradiction. Paradox of the main types of paradox, the probability of paradox, paradox geometry, statistics, and the time paradox paradox. Russell's paradox with its simple and clear shook the mathematical community, resulting in the third mathematical crisis. However, Russell's paradox is not the first paradox. Is not that old, shortly before Russell, Cantor, and 布拉里福蒂 have found contradictions in set theory. After the release of Russell's paradox, it appears a series of logical paradox. The paradox of the liar into the think of the ancient paradox. That is, Together, these paradoxes, causing a big problem, prompting everyone to care about how to resolve these paradoxes. The paradox was first published 布拉里福蒂 a paradox, the paradox is that the ordinal number according to their natural order form a well-ordered set. This well-ordered set, by definition, there is a ordinal Ω, Ω by the definition of the ordinal number should belong to the well-ordered set. But by the definition of ordinal numbers, ordinal sequence number of any section of the order within a period greater than any ordinal number, so should be better than any ordinal Ω are large, and thus does not belong to Ω. This is 布拉里福蒂 March 28, 1897 was read at the Ba Luomo mathematics in an article submitted. This is the first paradox of a modern released, it attracted interest in the mathematical community, and led to heated debate for many years after. Paradox has dozens of articles discussing the issue, which greatly promoted the re-examination based on set theory.布拉里福蒂 I think that this contradiction proves the natural order of the ordinal number is only a partial order,
vibram five finger, which Cantor proved that the results of a few months ago is a totally ordered set of ordinals in contradiction, and later 布拉里福蒂In this regard did not do the job. Russell in his French logician Jourdain found - a way out, he distinguished set of compatible and incompatible set. This distinction is actually Cantor has spent many years in private. Soon after, Russell in 1905 in an article set for the existence of ordinal raised questions about Zermelo have the same idea, then many people in this area are in possession of the same idea. Many classical mathematical paradox of the famous paradox of ancient and modern, they shook the foundation of logic and mathematics, and aroused the people to knowledge and sophisticated thinking, through the ages attracted the attention of many thinkers and lovers. Solve the paradox of problems require creative thinking to solve the paradox often can bring new ideas. This will cause the formation of paradox, it roughly classified into six types, at the upper, middle and lower three parts. This is the first part: that caused by the paradox of self-concept and brought the introduction of infinite paradox (a) caused by the paradox of self refers to the various cases of the following concepts are the existence of a self-referential or self-related question: If must start from the proposition, it will be negative proposition; if negative proposition from the start, will get its affirmative proposition. 1-1 paradox lies are the sixth century BC philosopher Democritus who Yipi Mini Andes (Epimenides): sources. This shows that the paradox is well known, but the logic of Paul to answer for it, and not interested. People will ask: Yipi Mini Andes are not lying? The simplest form of this paradox is the 1-2 Conflict is inevitable. It's a replica of 1-3 logical contradictions infinite loop. Topology of the single body is a figurative expression. Philosopher Bertrand Russell had seriously thought about this paradox, and trying to find a solution. In his come. This shows that something is faulty, but that is what is not correct. In the spring of 1903, including the discovery of a contradiction to the kind of logic I was enjoying a honeymoon interrupted. Paradox lies the most simple outline of the contradictions that he found: refers to the general what he said. just put into words included in the overall when it creates a paradox. those that do not involve the general proposition proposition; second level proposition is the general proposition of the first stage involves those propositions; the rest of this imitation, as well as infinite. and use logical terminology describing concepts, intended to avoid discrimination of natural language. But he says the preface of the book is: Then he pointed out the paradox in all logic has a easier to understand a point of view, if this paradox is what people say outside of Crete, the paradox will be automatically removed. But in set theory, the problem is not that simple. 1-4 Saville village barber paradox, the barber put up a sign: suddenly speechless. This is a contradiction reasoning: If the barber is not to their own hair, he is the kind of people sign on. Made it clear beforehand that he should give himself a haircut. Conversely, if the barber to his own hair, according to signs said,
timberland mesh boots, he only gave the village barber who do not give their own hair, he can not give yourself a haircut. Therefore, no matter how the barber answered, we can not rule out the inherent contradiction. The Russell paradox is raised in mid ○ 二年, so called This is the paradox of the popular set theory, a representation of the story. Obviously, there exists a non-excluded 1-5 Paradox of Set Theory If R contains itself, then, R does not belong to R. Following Russell's paradox of set theory mathematics problems found after the 1931 Godel (Kurt Godel ,1906-1978, the Czech people) proposed a system can be derived by the logic of This theorem states that: the system is not complete in any public setting, where there must be neither be denied nor affirmed the proposition. For example, Euclidean geometry Paradox 1-6 bibliography compiled a library of dictionaries a title, it lists all of the library is not listed in his book title. It does not list out the title yourself? This paradox is consistent with the barber paradox. 1-7 Socratic paradox of Adams and other famous sophists relative. He established the But his moral capacity of the Greeks is not, actually, when the seven-year-old complex that has been used as the representative of sophistry. Las expelled in Provincetown Tege, books were burned years later, Socrates was sentenced to death, but his theory was the inheritance of Plato and Aristotle. Socrates had a famous saying: . Ancient China also has a similar example: 1-7, Mohists later retorted: If We often say: 1-7 1-8 Paradox (paradox) from the Greek These examples illustrate that, logically, since they can not get rid of that brought the concept of a vicious circle. There is no further solution? In the following section will continue to explore the last part. (B) caused by the introduction of infinite paradox, On the. Achilles paradox 2-1 later in the ancient Greek mathematician Pythagoras, Zeno (Zeno of Elea), has made a number of well-known paradox of the future in mathematics, physics concept had a major impact, A Ji Lisi is one of paradox. Achilles (Achilles) is a good run in the Greek mythology hero. Zeno talk: Achilles can not catch up in the race started a little ahead of his tortoise, the tortoise because when he was starting to get to that point, the tortoise has moved forward to climb. The distance between Achilles and the tortoise can be infinitely close, but never catch the tortoise. Mr. Fang Lizhi had to use physical language to describe this problem: the paradox of Achilles used two different time metrics. General metric is: Suppose Achilles and the tortoise at the beginning of the distance S, the speed V1, respectively, and V2. When the time T = S / (V1-V2), Achilles to catch up with the tortoise. However, different methods of measurement Zeno: Achilles reaches the tortoise will be successive previous starting point, this time T '. For any T ', may be infinitely shorter, but Achilles is always behind the tortoise. The key is that T 'can not measure T = S / (V1-V2) a later time. 2-2 This is the dichotomy paradox Zeno's paradox: When a moving object for some distance to D, it must first reach one half the distance D, then the fourth one in eight, ten sixth, as well as can be infinitely divided down. Therefore, this object will never get to this D. These conclusions do not exist in practice, but logically impeccable. Zeno even think: T, catch a turtle, then, This means that the senses are not reliable, there is no logical and reliable. He said: According to the kinetic theory, Zeno also proposed a similar movement paradox :2-3 in this position and fixed there is no difference. Then, the sum of an infinite stationary position is equivalent to exercise it? Or is the movement of still infinitely repeated? Ancient China has a similar view, such as :2-4 This is the irresistible practical reasoning and conflict can not be avoided. German philosopher Friedrich Nietzsche in the Although Achilles start in the race to catch up completely in line with the facts leading turtle, but why Zeno used because of the Nietzsche said: In these two paradoxes, the If the infinite is not likely to be perfect, still not likely to become motion, then the truth is not the arrow flew, it did not shift, not out of the rest, the time has not passed. In other words, in this so-called, after all, only ######## reality, neither the time, space, and no movement. Finally, even the arrow itself is a virtual image, because it comes from diversity, freedom of the senses to evoke a vision of non-. Here is Nietzsche's analysis: assume that the arrow has a presence, then it is fixed, non-time, non-made while others, fixed and eternal. This is a ridiculous idea! It is assumed that movement is real, then the rest does not exist. Thus, the arrow has no place, no space. Is an absurd point of view! Assume that time is real, then it can not be infinitely divided. Arrow flight time required must be from a limited number of moments, of which each breath must be an atom. Is still a ridiculous idea! Nietzsche concluded that: all of our ideas, as long as its experience with, and learn the content from the visual world is treated as If there is absolute motion, there will be no space; If there is absolute space, there will be movement; if there is absolute, there will be diversity; if there is absolute diversity, there will be unity. In fact, the two paradoxes mentioned in this Kinematics of Newton, the start-up of calculus, but with no solid theoretical basis, there has been history of the The early nineteenth century, French scientists led by Cauchy established the limit theory, and later after the German mathematician Weierstrass further stringent, thus enabling the limit theory of a firm foundation for calculus, sports issues have been a reasonable explanation . I can see a limit in calculus and the theory of the invention or accepted before, it is difficult to explain the paradox of the movement. Different senses of thinking, when the Greeks used the concept of ruling the reality, if there are contradictions between the logic and reality, sensory Zeno accused of Can not find a reasonable explanation when the time thinking,
gel kinsei 3, intuitive form, a symbol or metaphor to no avail. Although Nietzsche's analysis of detailed, incisive, but he could not put them together. 2-5 Thousand years ago the ancient Chinese also used the concept of infinity. Song Hui Shi famous people Warring States (about BC 370 - before 310) the prime minister had Renliang Guo, Wizards argument is Chuang Tzu's friends, and Gong tied for the famous representative. His works have been Wang Yi and more, all from the other family only to see his words and deeds paper clips. Hui Shih emphasized the doctrine of all things were relative, and thus the difference between things is only a relative concept, existing and Huishi the strange proposition, for example, foot argument under the specific process. Hui Shih's paradox is also very influential in the West. From the perspective of Mao Zedong's basic acceptance dialectics Huishi infinitely divisible point of view. On August 18th *** In his conversation with the philosophy workers, said: electron itself have not yet been split, one day can be divided. 'foot of the hammer, on whichever half of the inexhaustible eternal', this is the truth. do not believe to try. If there is no science of the exhaust. It was noted that Mao is very partial to this sentence, such as the mid-fifties to the home Sanqiang, mid-August with the Zhou Peiyuan ***, Yu Guangyuan, 1973, 1974 met with Chen Ning Yang , Lee, etc., are mentioned in this sentence. ? 2-6 Twenty-year-old German mathematician Cantor Ph.D. (1845-1918) six years after the war to infinity. He successfully demonstrated: the points on a straight line and a plane that correspond to the points, but also a point in space and one to one. The infinite, within 1 cm-long line of points, and the Pacific surface points, and the whole point of the Earth's interior are However, the Cantor's Until 1897 the first International Mathematical Conference, and his recognition of the results was that almost all of mathematics to set theory based. Russell praised his work a mathematical basis, this is the Since then, mathematicians have made unremitting explored. For example, the 1996 Cambridge University Press published Heng Dika's It is mainly explained Heng Dika and mulberry flowers newly created IF (Independence-Friendly First-Order Logic) logic and its possible impact. It challenges many traditional concepts, such as the mathematical theory of axiomatic set theory as the appropriate framework for the Liar Paradox has made a further study. Whether it will lead to a logical and mathematical foundations of the revolution? We will wait and see. This is the second part: from a one-sided reasoning due to more fruit caused by the name real paradox and contrary to due to paradox. (C) by a one-sided due to more fruit this form of reasoning lead to the paradox of the paradox is similar to sophistry. Sophistry is disgusting in reality, but in the logic of a considerable position. Condorcet said: elegance, but it also exhausted their strength against the illusory problems. They are skeptical of natural philosophy, that the world is no absolute truths. Provincetown mentioned earlier Douglas (Protagras, about BC 485 - before 410) is a representative figure of his famous, he said: books. From Socrates to Aristotle are opposed to sophistry theory, Hegel said, Socrates often used to attack the sophistry of his school of dialectics, particularly Provincetown Douglas. Although many of these theories have lost a wise man, we can still Aristotle's According to records Aristotle, Plato (Plato, BC 427 - before 347) once said: sophistry is designed to discuss the For example, the And now every thing is often not who does it as that caused by the hereby (Paradox Solutions) concludes (ibid.). Exclusion of formal logic and dialectics of Hegel advocated (1770-1831) said that Plato invented the dialectics. Tak said: Where the existing things there must be the formation and disappearance of a process, and properties of things are not. However, occasionally we have to keep track of property as the essence of their reason; may therefore not available to understand why the academic establishment on property (In his view, sophistry, theory is Sophistry sound academic system, rather than knowledge. Condorcet in the Instead, a variety of systems in the casting; they ignore the facts of observation, as is well engaged in their own imagination;, since they can not put their own opinions on the basis of proof, it sophistry to try to maintain them. . The best way to deal with is the use of sophistry and dialectics to be verified in practice. 3-1 Why? Can not. Can too.相关的主题文章：
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