^{1}Grushka, Ya.I^{1}Institute of Mathematics of the NAS of Ukraine, Kyiv |

Dopov. Nac. akad. nauk Ukr. 2019, 8:9-15 |

https://doi.org/10.15407/dopovidi2019.08.009 |

Section: Mathematics |

Language: Ukrainian |

Abstract: The subject of this article is closely related to the theory of changeable sets. The mathematically rigorous theory of changeable sets was constructed in 2012. From an intuitive point of view, the changeable sets are sets of objects which, unlike elements of ordinary (static) sets, can be in the process of continuous transformations, i.e., they can change their properties, appear or disappear, and disintegrate into several parts or, conversely, several objects can merge into a single one. In addition, the picture of the evolution of a changeable set can depend on the method of observation (that is, on the reference frame). The main motivation for the introduction of changeable sets was the sixth Hilbert problem, that is, the problem of mathematically rigorous formulation of the fundamentals of theoretical physics. The notion of oriented set is the basic elementary concept of the theory of changeable sets. Oriented sets can be interpreted as the most primitive abstract models of sets of changing objects that evolve within a single (fixed) reference frame. The oriented sets are mathematical objects, in the framework of which one can give the mathematically rigorous definition of the concept of time as a certain mapping from a certain time scale, represented by a linearly ordered set, into the set of simultaneous states of the oriented set. In this paper, the necessary and sufficient condition of the existence of the onepoint time on an oriented set is established. From the intuitive point of view, the onepoint time is the time associated with the evolution of a system consisting of only one object (for example, one material point). Namely, the concept of a quasichain oriented set is introduced, and it is proved that the onepoint time exists on the oriented set, if and only if this oriented set is a quasichain. Using the obtained result, the problem of describing all possible images of linearly ordered sets is solved. This problem naturally arises in the theory of ordered sets. |

Keywords: changeable sets, ordered sets, oriented sets, time |

1. Gorban, A. N. (2018). Hilberts sixth problem: the endless road to rigour. Philos. Trans. A. Math. Phys. Eng. Sci., 376, No. 2118, 20170238. doi: https://doi.org/10.1098/rsta.2017.0238

2. Levich, A. P. (2009). Methodological difficulties in the way to understanding the phenomenon of time: new essences, formal means and criteria of understanding are needed. Time of the end of time. Time and eternity in modern culture (pp. 6688). Moscow: MoscowPetersburg Philosophical Club (in Russian).

3. Levich, A. P. (1995). Time as variability of natural systems: ways of quantitative description of changes and creation of changes by substantial flows. On the way to understanding the time phenomenon: the constructions of time in natural science. Part 1. Interdisciplinary time studies (pp. 149192). Singapore: World Scientific. doi: https://doi.org/10.1142/9789812832092_0010

4. Levich, A. P. (2009, January). Modeling of “dynamic sets”. Proceedings of the 5th AllRussian conference Irreversible processes in nature and technique (pp. 4346), Moscow (in Russian).

5. Barr, M., Mclarty, C. & Wells, C. (1986). Variable Set Theory. Cite Seerx. Retrieved from: http://www.math.mcgill.ca/barr/papers/vst.pdf

6. Bell, J. L. (2006). Abstract and variable sets in category theory. What is Category Theory? (pp. 916). Monza: Polimetrica.

7. Grushka, Ya. I. (2012). Changeable sets and their properties. Dopov. Nac. akad. nauk Ukr., No. 5, pp. 1218. (in Ukrainian).

8. Grushka, Ya. I. (2012). Primitive changeable sets and their properties. Bull. Shevchenko Sci. Soc., 9, pp. 5280 (in Ukrainian).

9. Grushka, Ya. I. (2013). Base changeable sets and mathematical simulation of the evolution of systems. Ukr. Math. J., 65, No. 9, pp. 11981218 (in Ukrainian). doi: https://doi.org/10.1007/s11253-014-0862-6

10. Grushka, Ya. I. (2012). Visibility in changeable sets. Transactions of Institute of Mathematics of NAS of Ukraine, 9, No. 2, pp. 122145 (in Ukrainian).

11. Grushka, Ya. I. (2017). Draft introduction to abstract kinematics. (Version 2.0). Preprint: viXra: 1701.0523v2. https://doi.org/10.13140/RG.2.2.28964.27521

12. Herrlich, H. (2006). Axiom of Choice. Berlin, Heidelberg: Springer. doi: https://doi.org/10.1007/11601562

13. Pincus, D. (1997). The dense linear ordering principle. J. Symb. Log., 62, No. 2, pp. 438456. doi: https://doi.org/10.2307/2275540