Material aspects of use of partial compensation of incompleteness of formal axiomatics

V. I. Bolshakov, V. Volchuk, Yu. Dubrov


In Gödel's incompleteness theorem proved that the theories constructed on the basis of a formal axiomatic meaning of the original terms and their interpretations are incomplete, due to the incompleteness of the language statements.

In this context, extending the conclusion of this theorem on the interpretation of statements of defining, for example, one of meanings the numerical value of the quality of any metal is accepted in meaning of incomplete statement .This incompleteness may be submitted by incomplete of constructial features of a formal axiomaticstructure (macro level), and the incompleteness of formal axiomatic at the micro level. The formal axiomatic of structural elements of the metal structure is obvious without requiring of further explanation. Because of it , the quality of the metal, is determined most of all on the basis of statistic or analysis or the history of its reception., at the current in



material science large number of studies aimed to determining of the quality of the metal, based on the analysis of its structure, because of computational irreducibility of resultate of this analysis,

S. Bir recommended to go beyond the initially selected language, but at the same time not to break away from the real situation, should be attached to such a property system that is inextricably linked to its real existence.

For partially compensate of the incompleteness of formal axiomatic metal structure is applicable language of much more higher level. From our perspective, such language is the language of fractal structures approximation of metal. This choice is based on the fact of a huge number of real physical systems possessing of (in the appropriate range scale) fractal nature, characterized by a fractional dimension. The concept of a fractal is almost due to both the characteristics of the metal structure and the physical characteristics produced of its products: with a rough surface; volume; density and others.

The received results show that the quality characteristics of the metal, is economically advisable to use, minimizing the number of actual tests, calculated with record of the fractal dimensions of its structure, i.


Godel's theorem; the metal structure; the theory of fractals; hardness; iron


Bir S. Kibernetika i upravleniye proizvodstvom [Cybernetics and production management]. Moscow, Nauka, 1963. 276 p. (in Russian). Available at:

Bol'shakov V. I. Fraktaly v materialovedenii [Fractals in material science]. Dnepropetrovsk, PGASA, 2006. 253 p. (in Russian).

Bol'shakov Vad. I., Bolshakov V. I., Dubrov Yu. I. Pro nepovnotu formalnoyi aksiomatyky v zadachakh identyfikatsiyi struktury metalu [About incompleteness of formal axiomatic in problems of identification of metal structure]. Visnyk Natsionalnoi akademii nauk Ukrainy − Bulletin of the National Academy of Sciences of Ukraine. 2014, no. 4, pp. 55-59. (in Ukrainian). Available at:

Gulyayev A. P. Metallovedeniye [Metallurgical]. Moscow, Metallurgiya, 1986. 544 p. (in Russian).

Dubrov Yu. I. Uchet vliyaniya neupravlyayemykh faktorov pri analize i sinteze kriteriya funktsionirovaniya slozhnykh sistem [Record for the effects of uncontrollable factors in the analysis and synthesis of criteria for the op- eration of complex systems]. Ekonomika i matematicheskie metody − Economics and Mathematical Methods. 1986, no. 1, pp. 165-170. (in Russian).

Klini S. K. Vvedeniye v matematiku [Introduction to Mathematics]. Moscow, IL, 1957. 527 p. (in Russian). Available at:

Saltykov S. A. Stereometricheskaya metallografiya [Stereometric metallography]. Moscow, Metallurgiya, 1976. 270 p. (in Russian).

Sinay Ya. G. Sluchaynost' nesluchaynogo [Chance of non - random]. Priroda. 1981, no .3, pp. 72-80. (in Russian).

Ivanova V. S., Balankin A.S., Bunin A. Zn. Sinergetika i fraktaly v materialovedenii [Synergy and fractals in mate- rial science]. Moscow, Nauka, 1994. 383 p. (in Russian).

Uspenskiy V. A. Teorema Godelya o nepolnote [Gödel's incompleteness theorem]. Moscow, Nauka, 1982. 110 p. (in Russian). Available at:

Mandelbrot B. B. The Fractal Geometry of Nature. Nev-Yuork, San Francisco, Freeman, 1982. 480 p.

GOST Style Citations

Бир С. Кибернетика и управление производством / С. Бир. – Москва : Наука, 1963. – 276 с. – Режим доступа:


Большаков В. И. Фракталы в материаловедении / В. И. Большаков, В. Н. Волчук, Ю. И. Дубров. – Днепропе- тровск : ПГАСА, 2005. – 253 с.


Большаков Вад. І. Про неповноту формальної аксіоматики  в  задачах  ідентифікації  структури  металу  / Вад. І. Большаков, В. І. Большаков, Ю. І. Дубров // Вісник Національної академії наук України. – 2014. -№ 4. – С. 55-59. – Режим доступу:


Гуляев А. П. Металловедение / А. П. Гуляев. – 6-е изд., перераб. и доп. – Москва : Металлургия, 1986. -544  с.


Дубров Ю. И. Учет влияния неуправляемых факторов при анализе и синтезе критерия функционирования сложных систем / Ю. И. Дубров, В. В. Фролов, А. Н. Вахнин // Экономика и математические методы. – 1986. – Т. 22, № 1. – С. 165-170.


Клини С. К. Введение  в  математику  /  С.  К.  Клини  ;  пер.  с  англ.  А.  С.  Есенина-Вольпина  ;  под  ред. В. А. Успенского. – Москва : Мир, 1957. – 527 с. – Режим доступа:


Салтыков С. А. Стереометрическая металлография (стереология металлических материалов) / С. А. Салтыков. – Москва : Металлургия, 1976. – 270 с.


Синай Я. Г. Случайность неслучайного / Я. Г. Синай // Природа. – 1981. – № 3. – C. 72-80.

Синергетика и фракталы в материаловедении / В. С. Иванова, А. С. Баланкин, И. Ж. Бунин, А. А. Оксогоев. - Москва : Наука, 1994. – 383 с.


Успенский В. А. Теорема Гёделя о неполноте / В. А. Успенский. – Москва : Наука, 1982. – 110 с. – (Популя- рные лекции по математике. Вып. 57). – Режим доступа:


Mandelbrot B. B. The Fractal Geometry of Nature / B. B. Mandelbrot. – New York ; San Francisco : Freeman and Company, 1982. – 480 p.