Kitabı oku: «All sciences. №1, 2023. International Scientific Journal», sayfa 3

Introduction

The basis of optoelectronic methods and devices are emitters and photodetectors. The widespread use of optoelectronic methods was hindered by the lack of simple reliable radiation sources. The appearance of semiconductor radiation sources has significantly expanded the scope of application of optoelectronic methods and devices [1].

At present, semiconductor emitters with a radiation spectrum ranging from the ultraviolet section to the near infrared section of the optical spectrum have been developed and are being mass-produced. Practically at present, it is possible to develop emitters in the range from 210 to 4000 microns with spectral characteristics close to monochromatic (with quasi-monochromatic spectral characteristics). The features of semiconductor emitters are high speed, the ability to control the radiation flow by current, monochromaticity, sufficient radiation power and small overall dimensions. The presence of such advantages in semiconductor emitters creates prerequisites for the research and development of various monitoring, measurement and conversion devices for various fields of science and technology. Hence follows a wide range of work in the field of creating devices and systems based on semiconductor emitters [2].

The basis of optical methods and devices is the presence of an emitter and an optically connected photodetector through the medium. The radiation generated by the emitter, passing through the medium (air, substance, etc.), is perceived by the photodetector. In these methods and devices, optical radiation is used as a data carrier that does not create electromagnetic interference and is not affected by these interference. The presence of such a feature and the simplicity of the instrument implementation create prerequisites for the research and development of various devices based on the use of optical radiation [3].

The main part

To build a moisture meter on semiconductor emitters, the property of water to absorb IR radiation of a certain wavelength is important [4]. All substances and materials have a certain hygroscopicity and, therefore, absorb moisture from the external environment. The analysis of spectral characteristics showed that the absorption bands lie in the range of 0.76…0.97 and 1.19…1.94 microns [5].

Table 1 shows the absorption spectra of water and their affiliation.

From the different spectral characteristics of dry matter (Fig. 1, curve 1) and at a humidity of 9% H2O (curve 2), it follows that at a wavelength of 1.94 microns, water has significant absorption [6]. In the humidity meter on semiconductor emitters, LEDs with radiation spectra of 2.2 microns are used as a reference channel, and LEDs with radiation spectra of 1.94 microns are used as a measuring channel).

Fig. 1. Emission spectra of LED1, LED2 LEDs and spectral sensitivity of PD24 photodiode.

LEDs based on the semiconductor compound GaSb and its solid solutions GaInAsSb and AlGaAsSb have been developed to measure the moisture content of raw cotton. LED structures are manufactured by the FEF method and grown on Gaas n-type conductivity substrates doped with Te to an electron concentration of 8·1017 cm-3. The emitters for measuring the moisture content of raw cotton consisted of an active layer of n – GaInAsSb (Eg = 0.51 eV) 2—3 microns thick and grown on n – GaSb substrates and doped with Te to a charge carrier concentration of 9·1017 cm—3, the wide-band emitter p – AlGaAsSb, doped with germanium to a concentration of 5·1018 cm-3 (fig.2).

Fig.2. GaSb-based LED for humidity measurement.

LEDs based on the GaSb semiconductor compound for measuring the humidity of raw cotton at a temperature of 24 0C had an external quantum photon output of 5.9 – 6.5% and an optical power of 3.9 MW in direct current.

To maximize the output of optical radiation, the TO-18 housing with a parabolic reflector is used, which allows collimating radiation at an angle of 10—11o. Figure 3 shows the design of the IR LED:

Fig. 3. LED with a parabolic reflector: a) design, b) radiation spectra, c) VAC (where: 1 – LED chip (1.94 microns), 2 – thermal cooler, 3 – LED chip (2.2 microns), 4 – parabolic reflector)

LEDs based on the GaAlAsSb/GaInAsSb/ GaAlAsSb double heterostructure had a quantum yield of 5.8%, a radiation wavelength of 1.94 microns for measuring the moisture content of raw cotton, its main parameters are shown in Table 2.

The proposed design provides equal conditions for two LED crystals, thus eliminating the time and temperature instabilities of their main parameters.

Fig. 3 shows a block diagram of a digital humidity meter, which consists of the following elements: a master generator – ZG; trigger – T; frequency divider – DC; differentiating devices – DU1, DU2; exponent modulator – ME; emitter repeater – EP; pulse amplifier – IU; radiation receiver – AF; low – noise amplifier – MSU; matching circuit – SS; counter – SCH; decoder – DS; indicator – IN; reference LED – ID1; measuring LED – ID2.

Fig.4. Block diagram of a humidity meter on semiconductor emitters

The characteristic features of the humidity meter on semiconductor emitters are high selectivity, sensitivity, accuracy and reproducibility of measurements, as well as the possibility of continuous non-destructive testing, contactless and rapid analysis.

Conclusion

To create a humidity meter on semiconductor emitters, the optimal absorption band free from the absorption band of interfering components l1 = 1.94 microns has been determined.

The optoelectronic device uses LEDs based on GaAlAsSb/GaInAsSb/GaAlAsSb (2.2 microns) as the emitting diode at the reference wavelength, and LEDs based on GaAlAsSb/GaInAsSb/GaAlAsSb (1.94 microns) as the emitting diode at the measuring wavelength.

The absolute error of the moisture content measurement results was 0.5%.

Literature

1. Bashkatov A. S., Mescherova D. N. «The main trends in the development of optoelectronic technology until 2030,» Abstracts of the Russian Conference and School of Young scientists on current problems of semiconductor photoelectronics «Photonics-2019», 2019, doi: 10.34077/rcsp2019—25. pp.25—26.

2. Bogdanovich M. V. «Water content meter in oil and petroleum products based on infrared optoelectronic LED-photodiode pairs,» Journal of Technical Physics, 2017, doi:10.21883/jtf.2017.02.44146.1791.

3. Masharipov Sh. M. Analysis of modern methods and technical means of measuring humidity of cotton materials. // Devices, 2016, No. 4., pp. 31—37.

4. Demyanchenko M. A. Absorption of infrared radiation in a multilayer bolometric structure with a thin metal absorber // Optical Journal. – 2017. Volume 84 – pp. 48—56.

5. Rakovics V., Imenkov A. N., Sherstnev V. V., Serebrennikova O. Yu., Ilyinskaya N. D., Yakovlev Yu. P. «Powerful LEDs based on InGaAsP/InP heterostructures,» fiz. i tekhnika poluprovodn., 2014.t.48.s.1693—1697.

6. Artemov V. G., Volkov A. A., Sysoev N. N. «The absorption spectrum of water as a reflection of charge diffusion // Izvestia of the Russian Academy of Sciences. Physical series, Proceedings of the Russian Academy of Sciences. The series is physical. – 2018. – vol.82. – pp. 67—71. doi: 10.7868/s0367676518010143.

DEVICES FOR REMOTE TEMPERATURE CONTROL BASED ON LEDS (λ=2.0 microns)

UDC 621.38

Qo’ldashov Obbozjon Xakimovich

Doctor of Technical Sciences, Professor of the Scientific Research Institute "Physics of Semiconductors and Microelectronics" at the National University of Uzbekistan
Ergashev Doniyor Jamoliddin ugli
2nd year Master of the Department of "Physics of Semiconductors and Polymers" of the Faculty of Physics of the Mirzo Ulugbek National University of Uzbekistan

Scientific Research Institute "Physics of Semiconductors and Microelectronics" at the National University of Uzbekistan

Annotation. An optoelectronic device for remote temperature control of small-sized objects is proposed, which can be successfully used in the study of temperature characteristics of solar installations.

Keywords: temperature, optoelectronics, sensor, control, LED, photodiode, block diagram, design.

Аннотация. Предложено оптоэлектронное устройство для дистанционного контроля температуры малоразмерных объектов, которое может быть успешно использовано при исследовании температурных характеристик гелиотехнических установок.

Ключевые слова: температура, оптоэлектроника, датчик, контроль, светодиод, фотодиод, блок схема, конструкция.

The device for remote temperature control contains a monitoring object 1, which through a modulator 2 is optically connected to the first radiation receiver 3, whose output is through the first amplifier 4, the first amplitude detector 5 and the first integrator 6, connected to the first input of the signal ratio receiving device 13, the second radiation receiver 7, whose output is through the second amplifier 8, the second amplitude detector 9 and the second integrator 10 are connected to the second input of the signal ratio receiving device 13 whose output is connected to the input of the recording device 14, the control device of the source of the collimated radiation 12, the input of which is connected to the output of the first amplifier 4, and the output is connected to the input of the source of the collimated radiation 11, which, through reflection from the surface of the controlled object 1, is optically connected to the second radiation receiver 7, an electric motor 15, the rotor of which is mechanically connected to the axis of rotation of the modulator 2. In Fig.4.13. the design of the modulator is shown. Here: 16 is the axis of rotation of the modulator; 17 is the modulating holes; 18 is a metal disk. Figure 4.14 shows time diagrams explaining the principle of operation of the proposed device. Figure 1 shows a block diagram, and Figure 2 shows the sensor design.

The optoelectronic device works as follows. The thermal radiation flux FPI1 (λ) of the control object 1, which is proportional to its temperature, passes the distance l, is modulated by the modulator 2 and enters the sensitive area of the first radiation receiver. The flux reaching the sensitive area of the first radiation receiver, according to the theory of optoelectronic devices, is defined as:

where: tc (λ) is the spectral transmittance of the atmosphere; Mco (λ) is the spectral density of the energy luminosity radiating from the surface of the controlled object; Ako is the area of the radiating surface of the controlled object; DP1 is the diameter of the entrance pupil of the first radiation receiver; l is the distance between the controlled object and the first photodetector.

Table 1 shows the main characteristics of photodiodes

Considering that

expressions (1) will take the form:

where: eko (λ) is the spectral coefficient of thermal radiation of the controlled object; MCHT (λ) is the spectral density of the energy luminosity of the black body. Considering that the radiation receiver operates in a limited spectral range, expression (2) for wavelengths λ1m, which corresponds to the maximum sensitivity of the first radiation receiver, can be written as:

where: ελ1mk0 is the spectral coefficient of thermal radiation of the controlled object at wavelengths λ1m; Mλ1mcht is the spectral density of the energy luminosity of the black body at wavelengths λ1m; τλ1mc is the transmittance of the atmosphere at wavelengths λ1m.

Fig.1. Block diagram of an optoelectronic device.

Fig.2. The design of the modulator.

Fig.3. Time diagrams of an optoelectronic device.

Fig.4. Sensor design.

Taking into account the Stefan-Boltzmann law that Mλ1mcht=σT4, expression (4) will take the form:

where: T is the temperature of the controlled object; σ=5.6697*10-8 W*m-2*K-4 is the Stefan-Boltzmann constant.

In addition, the sensitive area of the first radiation receiver 3 is affected by the heat flux of radiation from the modulator 2, which can be described by the ratio

where: ελ1mm0 is the spectral coefficient of thermal radiation of the modulator at wavelengths λ1m; Tmo is the temperature of the modulator; Amo is the area of the radiated surface of the modulator; lmo is the distance between the modulator and the first radiation receiver.

Therefore, the total flux acting on the sensitive area of the first radiation receiver has the form.

Then the output voltage of the first radiation receiver is defined as:

or

where: is the transmission coefficient of the first radiation receiver.

The voltage corresponding to expression (9) from the output of the second radiation receiver 3 is amplified by the first amplifier 4, as a result of which an alternating electrical signal is formed at its output (see Fig. 3.c) the amplitude of which is defined as:

where ky1 is the transmission coefficient of the first amplifier 4.

Since due to the use of a disk modulator with symmetrical modulating holes, the thermal radiation of the modulator itself, which affects the sensitive area of the first radiation receiver during the modulation period remains constant (see Figure 3a), i.e.

Therefore, the constant component of the total signal of the first radiation receiver 3 does not pass through the AC amplifier 4. That is, the amplitude of the variable component of the amplified signal is proportional only to the amplitude of the flux Fλ1mPI1.

The variable component of the amplified signal is detected by the first amplitude detector 5. The detected signal (see Figure 3.d) from the output of the first amplitude detector 5 is integrated by the first integrator 6 and fed to the first input of the signal ratio device 13.

In this case, the voltage supplied to the first input of the signal ratio receiving device 13, taking into account the above, can be described by the expression:

where k1=kPI1kU1kAD1kINT1 is the total transmission coefficient of blocks connected in series with the first radiation receiver 3, the first amplifier 4, the first amplitude detector 5 and the first integrator 6; kAD1 is the transmission coefficient of the first amplitude detector; kINT1 is the transmission coefficient of the first integrator.

When the output signal of the first amplifier 4 is exposed to the input of the control device of the collimated radiation source 12, an antiphase electrical signal is formed at its output. The latter is fed to the input of the collimated radiation source 11 and causes a pulsed flow of collimated radiation at its output.

The formed flow, by the source of collimated radiation 11, is induced to the area of the controlled object 1. In this case, the flow reaching the surface of the controlled object 1 in the case Ako ≤ Aki is defined as:

where Aki is the cross – sectional area of collimated radiation; τλ2mc is the transmittance of the atmosphere at wavelengths λ2m; Foλ2 is the initial flux of collimated radiation. In this case, the reflected flow from the surface of the controlled object 1 is defined as:

where uco is the reflection coefficient of the surface of the controlled object at wavelengths λ2.

In this case, the expression for the reflected modulated flux from the surface of the controlled object and reaching the sensitive area of the second radiation receiver 7 has the form:

where: DPI2 is the diameter of the entrance pupil of the second radiation receiver.

In addition, in the case of a partial coincidence of the radiation spectrum of the controlled object with the spectral sensitivity of the second radiation receiver 7, an unmodulated radiation flux from the controlled object at a wavelength of λ2m affects the sensitive area of the latter.

where: ελ2m is the spectral coefficient of thermal radiation of the controlled object at wavelengths λ2m;

Then the total radiation flux acting on the sensitive area of the second radiation receiver 7 has the form.

Therefore, the output voltage of the second radiation receiver is defined as:

or

where cFP2 is the transmission coefficient of the second radiation receiver.

The voltage corresponding to expression (18) from the output of the second radiation receiver 7 is amplified by the second amplifier 8, as a result of which an alternating electrical signal is formed at its output (see Fig.3. d) the amplitude of which is defined as:

where ky2 is the transmission coefficient of the second amplifier 8.

Since during the period the repetition of the modulation Uλ2mPI2 can be considered constant, i.e. (see Fig. 3.b)

Therefore, the constant component of the total signal of the second radiation receiver 7 does not pass through the AC amplifier 8. That is, the amplitude of the alternating component of the amplified signal is proportional only to the amplitude of the flux Fλ2mPI2.

The variable component of the amplified signal is detected by the second amplitude detector 9. The detected signal (see Figure 3. e) from the output of the second amplitude detector 9 is integrated by the second integrator 10 and fed to the second input of the signal ratio device 13.

In this case, the voltage supplied to the second input of the signal ratio receiving device 13, taking into account the above, can be defined as:

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).