Gödel's theorem of Gödel's Theorem
mathematical logic is a theorem, the logic of Austria in 1931, Celtic mathematician Gödel (Kurt Godel) to discover and prove, this theorem completely shattered in the form of Hilbert ideals.
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theorem and the significance of application of theorem and proof of this theorem and its significance for understanding the significance, requires considerable knowledge of mathematical logic and set theory. Make all these preliminaries are sorted out here, working too hard, and this is what I have dared to write this thing hands one of the reasons. Still not going to detail here, these things only when necessary to give some simple instructions, in order to gain a deeper understanding of friends who are interested can study relevant courses. Gödel's theorems are actually two theorems, in which Gödel's first incompleteness theorem is the most important and most misunderstood, from the many versions of this theorem can be seen. Such as:
defined the concept of natural numbers,
nike shoes, we can construct the system which can neither prove nor can permit the proposition. A corollary: The proof is not easy to understand even the. I steal lazy, skip directly to introduce the significance of it. Gödel's theorem is the first-order logic theorem, in the form of logic, mathematical proposition and its proof are described in a symbolic language, where we can mechanically check the proof of the legitimacy of each, so they can be from a group of Start irrefutable proof of an axiom theorem. The last century, as represented by Hilbert formalism camp, hope that through the method of formal logic, to construct a relevant number theory (natural number) of the finite set of axioms, the introduction of all the principles of number theory (completeness), and no contradiction ( compatibility), and as the starting construct a mathematical system of the whole formalism. The first incompleteness theorem Gödel, crushed the idea. In fact these two theorems show that this axiom system are either incomplete or contradictory. On the compatibility of the number of root Heights (G. Gentaen ,1909-1945) in 1936, contains non-use of deductive logic proved by transfinite induction. Thus,
nike shoes, the theorem reveals, in most cases, such as number theory or real analysis can never find the complete set of axioms. You can keep adding new axioms axiom system, and even constitute an infinite set of axioms,
nike shoes, but this is no longer a recursive list of axioms, there is no mechanical method to judge whether the judge is the added axiom axiom axiom systems. This is of great significance in computer science, computer languages, the first-order logic theorem is recursively enumerable, however Gödel's first incompleteness theorem shows that such programs can not be prepared, through the recursion theorem proving, for a limited time true and false propositions within the judge. Penrose's Of course, this point is not conclusive, there is much controversy. Some say the simple, but it seems not. Or a combination of the analysis of common misconceptions and try to clarify the vague understanding of it. A common misconception: This is the most common, and some even use this to deny the logic, this is wrong. Take Euclidean geometry, it can be axiomatized as a complete system. Two common misconceptions: Some of this error from the above description of Gödel theorem can see that. This theorem axiom system can only assume that Many systems contain natural numbers, such as Three common misconceptions: No, we can prove by other means, such as mentioned above, the transfinite induction. In fact,
nike shoes, the theorem only shows that we can not demonstrate compliance within the system, we can not rule out other systems is proved. For example, the Peano axioms of number theory alone can not prove that the range in number theory, but can be proved in set theory. Gödel's Theorem applied mainly in the application of mathematics, as the use in practice, many are based on an incorrect understanding of the blind to apply, the forum some of the missionary post in there on many occasions. To avoid mistakes, or should understand the real meaning of this theorem, but, paradoxically, it does require considerable mathematical knowledge-based, it may be, for many people, very difficult. So I pointed out some errors in the text to understand,
nike air Cashier stock accounting of public funds to buy mil, for a simple analysis, if we have to pay attention to avoid many mistakes. Now I introduce some examples of practical application, but I personally think that the application of more accurate, may help us to understand the theorem of help, since it is a mathematical theorem, the most direct applications or in mathematics, particularly number theory. A lot of number theory proof of the proposition,
nike shoes, we need to use Gödel's theorem. This I not introduced. The theorem shows that some propositions about the natural numbers themselves may be true proposition, but not only from the justice system proved a natural number or evidence not, the need for other means or methods, such as set theory, before can attest. It has been speculated, methods to determine which system is the proposition in which the permit. Therefore, even for very abstract and formal mathematics, mathematician's intuition - that is, the accumulation of a large number of practical experience - pure forms of mathematics than the more basic logical reasoning, and more reliable. But it does not say this logic is useless, which can be used to verify that the former is correct,
nike shoes, you can deduce some new propositions right,
nike shoes His mother ran away from home more than, but not representative of all. And as previously mentioned, if not proved in the form of the system, but also through other methods, or from other systems in the proof. In addition, once again stressed that This frequently misunderstood, it is recommended friends who are interested to know more about master the basics. The theorem is another major application fields of application is a branch of mathematics - computer and artificial intelligence. Now I mentioned the text issues a brief shutdown. Now, with a great computer to the development, but the basic principle or out of Von Linus, but the speed and efficiency is greatly increased. Basically, the computer program is a binary digital computing based on 2 propositional calculus system. One axiom is limited given the rule is calculated, and determine the authenticity of the proposition, the output results,
air max for cheap Killed one police official said, turn to the next stop and deal with a proposition. This is the first in line with the Godel incompleteness theorem conditions. If the theorem can be said, such a system is necessarily incomplete, that proposition can not at least one such As the saying goes is to be Expansion of justice no matter what you set, as long as is finite, this phenomenon will persist. The infinite set of axioms for the computer,
nike shoes, it means infinite storage space, which is obviously impossible. Therefore, some mathematicians, Penrose, as I mentioned to believe that this shows that computers have a fatal flaw, and a human AI truly hope that in the wisdom of the Details on this issue, read Penrose's Why does the human brain and computer have such a fundamental difference between it, Penrose that quantum mechanics may be uncertainty and complexity of the role of non-linear chaotic system jointly caused. But some mathematicians do not think so, they noted that the basic meaning and the human brain works on it, and the artificial intelligence principle Gödel's theorem should be limited. The difference between the two, can include non-deterministic computing system that is the so-called Contains the human brain is such a natural form of non-deterministic neural network system, The reason it looks with the computer does not have the . The traditional deterministic Turing machine is a serial processing system, although it can simulate this kind of While working on the quantum computer and the computer neural network system only real hope to solve this problem, the human brain's ability to achieve. The computer is a I do not have the ability to determine who is right, but I personally think that Various studies have shown that the human brain in the the original of the Gödel's theorem and the conditions that no essential differences. The understanding of the process of Take the language (that form), it can be converted into a limited axioms and rules of symbolic logic systems is a consistent axiom system in the form of Theorem conditions. This theorem precisely that,
nike shoes, such a system is not perfect, there can not be confirmed with the proposition of the system, for this system,
nike shoes, it is the language of the expression of thought is not complete, that is, we often say, . This also we often feel There is another fact that this point is the translation. The form of text language translation of the text though not difficult, but the original language to express accurately the exact meanings on very difficult, even impossible things can be said. If you can prove that human thinking can be turned into axioms in such a way that the human brain can be restricted by the Gödel theorem. Argument when applied to practical problems, the Gödel theorem - any logic system,
nike shoes, which itself must be limited or is self-contradictory - that would explain the, when we see the world,
nike shoes, in fact, must be used alternately with limited systems (formal logic) or is it to look at the dialectics. Because the thing itself exist only limited and there was a contradiction, we will inevitably come to the conclusion that we how to apply. If we limited the level of use of conflicting views, then we will both be closed and painful; level if we are in contradiction with a limited point of view, it is the truth to say dead. And so on. Atlas Atlas entry for more extended reading: 1
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