All sciences. №1, 2023. International Scientific Journal

where k2=cFP2kU2kAD2kINT2 is the total transmission coefficient of the blocks connected in series of the second radiation receiver 7, the second amplifier 8, the second amplitude detector 9 and the second integrator 10; kAD2 is the transmission coefficient of the second amplitude detector; kINT2 is the transmission coefficient of the second integrator.

It is known that optical devices designed to measure temperature mainly use a transparent region of the atmosphere spectrum. Therefore, for a small distance between the object of control and the radiation receiver, it can be assumed that, τλ1mc=τλ2mc"1. Then, when using identical electronic blocks for the radiation fluxes Fλ1mPI1 and Fλ2mPI2, we have k1 = k2. Therefore, at the output of the signal ratio receiving device 13, in proportion to the temperature of the control object 1, a voltage ratio is formed:

or

Since solar parabolocylindrical concentrators have a reflection coefficient in the near and middle IR spectral region that is constant and is γλ2ko = 0.1.

Then the temperature in the local focal zone of solar parabolocylindrical concentrators is defined as:

Thus, it can be seen from the last expression that the temperature in the local focal zone of solar parabolocylindrical concentrators is proportional to the voltage ratio Uλ1m and Uλ2m, which is recorded by the recording device, where it is taken into account.

Literature

1. Ergashev S. F., Kuldashov O. H. Control of gas concentration in geothermal energy. NTZH FerPI, 2014.No.3. from 105-109.

2. Daliev S. H., Nasriddinov S. S., Kuldashov O. H. The use of LEDs (1.94 µm) to measure the moisture content of raw cotton. Proceedings of the International conference "Optical and photoelectric phenomena in semiconductor micro- and nanostructures". Fergana, 2020, pp.426—427.

3. Kuldashov O. H. Optoelectronic device for remote temperature control of raw cotton riots. International Conference "Geoinformation support of aerospace monitoring of hazardous natural processes". Irkutsk, NIU, 2010.

4. Bezylaznaya T. V., Bogdanovich M. V., Kabanov V. V., Kabanov D. M., Lebedok E. V., Paraschuk V. V., Ryabtsev A. G., Ryabtsev G. I., Shpak P. V., Shchemelev M. A., Andreev I. A., Kunitsyna E. V., Sherstnev V. V., Yakovlev Yu. P. Optoelectronic pairs of LEDs-photodiode based on the InAs/InAsSb/InAsSbP heterostructure for carbon dioxide detection. Physics and Technology of Semiconductors, 2015, volume 49, vol. 7. C1003—1006.

5. Jha S. et al.«Violet-blue LEDs based on p-GaN/n-ZnO nanorods and their stability // Nanotechnology. – 2011, doi: 10.1088/0957—4484/22/24/245202.

MATHEMATICAL SCIENCES

THE PARADOXES OF MATHEMATICS POPULAR IN MODERN SCIENCE

UDC 520.254

Aliyev Ibratjon Xatamovich

2nd year student of the Faculty of Mathematics and Computer Science of Fergana State University

Aripova Sayyora Boxodirovna

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, as well as countable and uncountable sets. And to admit, this work was not a bad stress for mathematicians of that time, because for 2000 years Euclidean geometry, which was considered ideal, was going through difficult times thanks to Lobachevsky and Gauss, who discovered non-Euclidean geometry, this led to a poor definition of the limit – the basics of mathematical analysis.

And now Mr. Kantor has decided to contribute to these processes, showing that infinity is much more complicated than it seemed. Because of this, no small disputes broke out, dividing mathematicians into 2 camps – intuitionists who believed that Cantor's work was nightmarish, and mathematics was an invention of the human mind, and Cantor's infinities could not simply be. Unfortunately, Henri Poincare, who wrote: "Posterity will read about set theory as a disease that they managed to overcome," and Leopold Kroniker called Cantor a charlatan scientist and a corrupter of young minds. And also diligently interfered with his career.

They were opposed by formalists who believed that set theory would put mathematics on a purely logical basis. And their non-official leader was the German mathematician David Hilbert, who at that time became a living legend, with works in almost all areas of mathematics, having created concepts that became the basis of quantum mechanics, and he knew perfectly well that Cantor's work was brilliant. After all, such an idea, a strict and clear proof system based on set theory could solve all mathematical difficulties, and many agreed with him. This is also proved by his words: "No one can expel us from the Paradise that Kantor created."

But in 1901 Bertrand Russell pointed out a serious problem in set theory, because if a set can contain anything, it also contains other sets and even itself. For example, the set of all sets must contain itself, as well as the set of sets with more than 5 or 6 elements, or the set of all sets containing themselves. And if you accept this, it turns out to be a strange problem, because what to do with the set of all sets that do not contain themselves?

After all, if this set does not contain itself, it must contain itself, and if it does not contain itself, then by definition, it must contain itself. It turns out the paradox of self-reference, where the set contains itself only if it does not contain itself and does not contain itself only when it contains. But his allegory is more popular, with a city where only men live and a barber should shave only those men who do not shave themselves, but the barber himself is also a man and lives there. But if he does not shave himself, then a barber should shave him, but he cannot shave himself, because he does not shave those who shave themselves, it turns out that he should shave himself only if he does not shave himself. And of course, intuitionists were happy about this paradox.

But Hilbert's followers solved this problem by simply changing the definition to the fact that the set of all sets is not a set, just like a set of sets that does not contain itself. And although the "battle" was won, the self-reference remained and awaited its revenge.

This problem has been revived since the 60s of the XX century, when mathematician Hao Wang was thinking about ways to decompose multi-colored tiles by setting the following conditions – you can combine the edges of the same color, but you cannot rotate or flip the cell. And then the question arises, is it possible to tell from a random set of tiles whether it is possible to pave the entire plane? Is it possible to do this indefinitely and surprisingly, this task has become unsolvable, like the game "Life" and the whole problem has again been reduced to the already familiar self-reference, which has yet to be learned.

And then Hilbert decided to create a reliable proof system. The main idea of such a model was back in ancient Greece, where some initial statement was taken for truth without evidence – an axiom, for example, that only one straight line can be drawn between two points and evidence from consequences is built on the basis of these statements. So it turns out to preserve the truth of statements, where if the original ones are true, the new ones are also true.

So Hilbert wanted to get a symbol system – a language with a strict set of operations, where mathematical and logical statements could be translated into this language, and the phrase if you drop the book, it will fall down to (1).

Which read: «If A, then B.» And the statement that «There are no immortal people» would look like (2).

So the formalists wanted to give mathematical axioms the form of symbolic statements and establish the rule of inference as mathematical operations in this system. Russell, together with Whitehead, decomposed and described such a formal system in the three-volume "Principles of Mathematics", published in 1913, which became a monumental work of 2,000 pages of dense mathematical text, where on 762 pages a proof is given that 1 +1 =2, after which it is stated that "the above application sometimes turns out to be useful" ("The above proposition is occasionally useful"). They planned to write the 4th volume, but it seems fate did not like it, speaking more figuratively and giving a not bad example.

The thing is that although such mathematical records are too unusual, they are brief and accurate than ordinary language, leaving no room for errors or fuzzy logic, allowing you to describe the properties of the formal system itself. And if such an opportunity has finally appeared, then this is the time to study mathematics itself, posing three main questions:

1. Completeness of mathematics, that is, is it possible to prove any true statement?

2. Consistency, that is, is mathematics free from contradictions? After all, if you can say that A is true and that A is false at the same time, then you can prove anything and all meaning in science itself disappears.