Figure 6 (a) It is suggested by Axiom 2 that the space-time extension of a velocity can reach infinite distance, the ratio of the velocity is arbitrary and either finite or infinite. (b) It is known from Axiom 3 that if two inertial systems are compared for finite quantities, then the extension of the quantities of the two inertial systems must remain within the finite range and do not reach infinite distance. (c) If two inertial systems are infinite versus finite, then it is known from Axiom 3 that a change of direction means it is infinitely great and the finite is not part of infinitely great, so this extension of infinitely great is defined to be inextensible.
 
Because Axiom 3 is a modification of Axiom 1 (that is, Axiom 3 retains some of the properties of Axiom 1), if two inertial systems involve a comparison of finite values, then these two inertial systems extend only in a finite range and cannot extend to infinity (derived from Axiom 3). If the two inertial systems are unlimited (i.e., infinite) compared with limited amount of, so learn from Axiom 3, in which the direction of change means infinitely great , and the finity is not part of infinitely great, that then for infinity (infinitely great) has two meanings. Firstly, it is the largest unit (with an infinitely great unit), i.e., there is no bigger or smaller amount, and therefore this extension of infinitely great is defined as inextensible (Figure 6). Secondly, the change in direction means that it cannot be added, subtracted, multiplied or divided, and that it is not a finite component, so it does not vary with the corresponding value of a finite number. Therefore, the Lorentz transformation in the two inertial systems of relativity and the modified Lorentz transformation (corresponding to changes in time and space length), or other magnitude and value transformations (which apply to Axioms 1 and 2), are meaningless in Axiom 3. Instead of the spatiotemporal coordinate transformation or numerical transformation of the two inertial systems defined in Axioms 1 and 2 (only in the motion of uniform linear velocity), the spatiotemporal transformation of the two different inertial systems in Axiom 3 only changes in one direction, which is a unique quantity-value transformation and represents all quantity-value transformations (not only in the motion of uniform linear velocity but also in non-uniform linear velocity).
 
7 Conclusions
 
(1) It is concluded from Axiom 1 that a definition of velocity in relativity that consists of two dimensions representing the relationship between space and time is not valid and there is only independent one-dimensional space or time in Axiom 1. As a result, the principle of relativity and the principle of the constant velocity of light are substituted by the principle of the inertial system of Axiom 1 and the principle of universal invariant velocity of Axiom 1.
(2) Unlike two dimensions whose magnitudes of space and time are determined by the ratio between the two, the magnitude of a single dimension is determined by the unit values of one dimension, which indicates that any velocity (including infinitely great velocity) is meaningless and there is only infinitely great space in one dimension and infinitely long time in one dimension.
(3) Because Axiom 3 is a modification of Axiom 1, it retains some properties of Axiom 1 despite its new properties . Unlike Axiom 1, in which the transition from finite to infinite is a continuous process, in Axiom 3, the transition from finite to infinite involves a leap, thus, if the extensions are within the range of finite quantities for two inertial systems in Axiom 3, they must only stay in the finite range and do not reach infinite distance. If these two inertial systems are infinite versus finite, then it is known from Axiom 3 that the change in direction means infinite great ,and this extension of infinite great can be defined as inextensible.
There are some limitations for this study presently. First, there have been no direct observations made to confirm the conclusions of this study. Second, due to the difficulty of observing infinity, this research rests only on logical reasoning, but this does not prevent it from redefining or approximating the relationship between the physical quantities in the observable finite range of time and space. In other words, the conclusion is still applicable to physical quantities within this range. Thus, one of the greatest benefits of this study may be that we can redefine the mass–energy equation.
 
8 Prospects  
 
This paper discusses the concept of inertial systems in Axiom 1 (i.e. uniform linear motion), so the reader may ask, how does Axiom 1 define the concept of non-inertial systems (e.g., acceleration or curved motion)? Because two dimensions do not exist in Axiom 1, neither do many dimensions, so how does a single dimension define a non-inertial system (e.g., acceleration)? I will discuss this issue in detail in my next paper.
 
                    
References
[1] G. Amelino-Camelia, “Relativity in spacetimes with short-distance structure governed by an observer-independent (Planckian) length scale” Int. J. Mod. Phys. D., vol. 11, pp. 35–59, 2002.
[2] M.K. Iqbal, M. Abbas, and I. Wasim, “New cubic B-spline approximation for solving third order Emden–Flower type equations,” Appl .Math. Comput., vol. 331, pp. 319-333, 2018.
[3] N. Khalid, M. Abbas, and M. K. Iqbal, “Non-polynomial quintic spline for solving fourth-order fractional boundary value problems involving product terms,” Appl. Math. Comput., vol.349, pp. 393-407, 2019.
[4] N. Khalid, M. Abbas, M. K. Iqbal, J. Singh, and A.I.M. Ismail, “A computational approach for solving time fractional differential equation via spline functions,” Alex. Eng. J., 2020, in press
[5] T. Akram, M. Abbas, M. B. Riaz, A. I. Ismail, and N. M. Ali, “An efficient numerical technique for solving time fractional Burgers equation,” Alex. Eng. J., vol. 59, pp. 2201-2220, 2020.
[6] Qing Li. A geometry consisting of singularities containing only integers. (preprint Research Square: DOI: 10.21203/rs.3.rs-219046/v1 )
[7] Qing Li, The meaning of the infinitely great  (preprint author: DOI: 10.22541/au.160822935.50569408/v1)
[8] A. Einstein. “The meaning of relativity,” Beijing Science Press. pp. 22-23 (1979)
 
The data availability statement:
The [DATA TYPE] data used to support the findings of this study are included within the article.
 
Author information:
Qing Li
Code Number:050031
402, Unit1, Building 28
West zone of ChangRong Small District
No. 122,YuHua East Road DongYuan Street
YuHua District
ShiJiaZhuang City HeBei Province PR. China. Tel.: +86-13833450232 E-mail: liqingliyang@126.com
Backup e-mail: 2895621512@qq.com
Author contributions: Qing Li completed this manuscript in full.
Funding Acknowledgement: This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.
Conflict of interest statement: The author declares no conflict of interest in preparing this article.