发表机构
HUN-REN Alfréd Rényi Institute of Mathematics; School of Computer Science, The University of Sheffield; University of Public Service（匈牙利科学院阿尔弗雷德·雷尼数学研究所; 谢菲尔德大学计算机学院; 公共服务大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文在任意有序域上重构Borisov公理系统，证明非阿基米德域上存在欧几里得、伽利略和庞加莱模型及无限下降链，且欧几里得等距变换无法被自然公理消除。
AI 中文摘要
1978年，Yu. F. Borisov提出了一个公理系统，该系统使用几个基本假设和四条显式公理，其中第四条是相对性原理的表述；他证明了该公理系统（在单位选择的意义上）只有两个模型：一个相对论模型，其中世界观变换是庞加莱变换；一个经典模型，其中世界观变换是伽利略变换。在本文中，我们在一个直观简单但严格形式化的一阶逻辑框架内重新表述Borisov的原始四条公理，并将其基本背景假设转化为显式公理。我们不假设物理量的结构是实数域，而只假设它们构成一个有序域。这使我们能够研究Borisov定理如何依赖于量的结构。我们证明（作为我们的主要贡献）如何在每个非阿基米德域上构造Borisov公理系统的欧几里得、伽利略和庞加莱模型。我们还证明了在这三种情况下都存在一个无限下降的模型链和变换群链，这在阿基米德域上是不可能的。作为一个应用，我们注意到存在一个满足相对性原理的Borisov公理模型，其中世界观变换是欧几里得等距变换。在实数域上，利用关于时间箭头和不存在瞬时运动的自然公理，很容易消除这个模型。然而，在非阿基米德域的情况下，欧几里得等距变换作为Borisov公理模型中的世界观变换内在地出现，并且无论是时间箭头的假设还是对瞬时运动的拒绝，都不能消除它们。
英文摘要
In 1978, Yu. F. Borisov presented an axiom system using a few basic assumptions and four explicit axioms, the fourth being a formulation of the relativity principle; and he demonstrated that this axiom system had (up to choice of units) only two models: a relativistic one in which worldview transformations are Poincaré transformations and a classical one in which they are Galilean. In this paper, we reformulate Borisov's original four axioms within an intuitively simple, but strictly formal, first-order logic framework, and convert his basic background assumptions into explicit axioms. Instead of assuming that the structure of physical quantities is the field of real numbers, we assume only that they form an ordered field. This allows us to investigate how Borisov's theorem depends on the structure of quantities. We demonstrate (as our main contribution) how to construct Euclidean, Galilean, and Poincaré models of Borisov's axiom system over every non-Archimedean field. We also demonstrate the existence of an infinite descending chain of models and transformation groups in each of these three cases, something that is not possible over Archimedean fields. As an application, we note that there is a model of Borisov's axioms that satisfies the relativity principle, and in which the worldview transformations are Euclidean isometries. Over the field of reals it is easy to eliminate this model using natural axioms concerning time's arrow and the absence of instantaneous motion. In the case of non-Archimedean fields, however, the Euclidean isometries appear intrinsically as worldview transformations in models of Borisov's axioms and neither the assumption of time's arrow, nor the rejection of instantaneous motion, can eliminate them.
Comments28 pages, 5 figures
Journal refJudit X. Madarasz, Mike Stannett, Gergely Szekely, Groups of Worldview Transformations Implied by Einstein's Special Principle of Relativity over Arbitrary Ordered Fields. The Review of Symbolic Logic. 2022;15(2):334-361
