LITMIR.BIZ

      Teacher of secondary school No. 1 of the city of Ferghana

      Annotation. There is a weak spot in the foundation of mathematics, because of which it is impossible to know everything for sure, there will always be true statements that cannot be proved, no one knows exactly what these statements are, but they are similar to the hypothesis of «twin numbers». So pairs of prime numbers, where one of them is larger than the other by 2, for example 11 and 13 or 17 and 19. If you go higher up the numerical line, prime numbers are becoming rarer, not to mention such pairs. But the hypothesis about prime numbers says that there are infinitely many of them. So far, no one has been able to prove or disprove this yet.

      Keywords: mathematics, calculations, discrete mathematics, logic.

      Аннотация. В фундаменте математики есть слабое место, из-за чего нельзя знать всё наверняка, всегда будут истинные утверждения, которые нельзя доказать, никто точно не знает, что это за утверждения, но они похожи на гипотезу о «числах близнецах». Так пары простых чисел, где одна из них больше другого на 2, например 11 и 13 или 17 и 19. Если идти выше по числовой прямой простые числа встречаются всё реже, не говоря уже о таких парах. Но гипотеза о простых числах гласит, что их бесконечно много. До сих пор никто ещё не смог это доказать или опровергнуть.

      Ключевые слова: математика, расчёты, дискретная математика, логика.

      But the amazing thing is that most likely no one will ever be able to do it. After all, it is well known that in any mathematical system where operations are defined, there will always be true statements that cannot be proved. The best example is the mathematical model of the game "Life", created by mathematician John Conway in 1970.

      "Life" unfolds on an endless field of square cells, each of which is either "alive" or "dead", there are only 2 rules in the game: any dead cell with 3 neighbors comes to life and any living cell with less than 2 or more than 3 neighbors dies. So you can set the initial configuration of the location of points and the model creates the first, second, third and subsequent generations. Everything happens automatically, although the rules are simple, they generate quite complex behavior, where the following situations arise:

      1. Stable states that freeze in place;

      2. Looping in an endless loop, constantly flickering;

      3. They run away in an endless field, like gliders;

      4. Simply mutually destroyed;

      5. Living forever and creating new cells.

      And looking at such conditions, I would like to assume that any behavior can be predicted, whether they will come to rest or will grow indefinitely depending on the initial conditions. But no matter how strange it may be, it is not possible to do this. That is, it is impossible to create an algorithm that would find the answer in a finite period of time, without executing the algorithm itself, up to a certain point, but even so, it is possible to talk only about the final account of time, that is, up to a certain number of generations, and not about infinity.

      But what is even more surprising is that such unsolvable systems are not isolated and obviously not rare. You can bring Wang tiles, quantum physics, air ticket sales or card games. But to understand how the unsolvability arises in these cases, we will have to go back to the times of the XIX century, when this split happened in mathematics.

      In 1874, the German mathematician Georg Kantor published his work, giving rise to "Set Theory". Sets are an accurately described collection of something, which can include anything – shoes, planetariums of the world, people. But among such sets there are also empty ones – there is simply nothing in them, but there are also sets containing absolutely everything – these are universal sets.

      But Cantor was not interested in so many things, but in so many numbers, namely, the sets of natural numbers are all integers, rational numbers are all numbers that can be represented as fractions, this also includes integers, as well as those included in the set of rational – the set of irrational numbers – the number "pi", Euler, the root of two, as well as any other number that can be represented as an infinite decimal fraction. Cantor's question was to determine which numbers are greater – natural or real in the range from 0 to 1. On the one hand, the answer seems obvious – both are infinite, that is, the sets are equal, but some table was created to demonstrate this.

      The idea of the table is extremely simple – let each natural number correspond to a certain real number in the range from 0 to 1. But since these are infinite decimals, they can be written in random order, but the most important thing is that absolutely everything is present and there is not a single repetition. If, as a result, there are no extra numbers left when checking with a certain super machine, then it turned out that the sets are the same.

      And even if we assume that this is the case, Cantor suggests inventing another real number as follows. He adds one to the first digit after the decimal point of the first number, then one to the second digit of the second number, one to the third digit of the third number, etc., if 9 comes across, subtract one, and the resulting number is still in the same interval between 0 and 1, while never repeating itself in the whole list, because from the first numbers it differs from the first, from the second by the second, from the third by the third, etc. by numbers up to the very end.

      That is, it differs from each number by at least one diagonal digit, hence the name – Cantor's Diagonal Method, which proves that there are more rational numbers between 0 and 1 than all natural ones. It turns out that infinities can be different, hence the concepts of continuum,

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