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GROUPS OF WORLDVIEW TRANSFORMATIONS IMPLIED BY EINSTEIN’S SPECIAL PRINCIPLE OF RELATIVITY OVER ARBITRARY ORDERED FIELDS

Part of: Special relativity Linear algebraic groups and related topics General logic

Published online by Cambridge University Press:  23 March 2021

JUDIT X. MADARÁSZ ,
MIKE STANNETT  and
GERGELY SZÉKELY
JUDIT X. MADARÁSZ
Affiliation:
DEPARTMENT OF SET THEORY, LOGIC AND TOPOLOGY ALFRÉD RÉNYI INSTITUTE OF MATHEMATICS P.O. BOX 127, BUDAPEST 1364, HUNGARYE-mail: madarasz.judit@renyi.huURL: http://www.renyi.hu/~madarasz
MIKE STANNETT
Affiliation:
DEPARTMENT OF COMPUTER SCIENCE THE UNIVERSITY OF SHEFFIELD PORTOBELLO, SHEFFIELD S1 4DP, UKE-mail: m.stannett@sheffield.ac.ukURL: https://www.sheffield.ac.uk/dcs/people/academic/mike-stannett
GERGELY SZÉKELY
Affiliation:
ALFRÉD RÉNYI INSTITUTE OF MATHEMATICS P.O. BOX 127, BUDAPEST 1364, HUNGARY and DEPARTMENT OF NATURAL SCIENCE UNIVERSITY OF PUBLIC SERVICE P.O. BOX 60, BUDAPEST 1441, HUNGARYE-mail: szekely.gergely@renyi.huURL: https://users.renyi.hu/~turms
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Abstract

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.

Keywords

first-order logicrelativity theoryclassical spacetimespecial principle of relativitytransformation groupsaxiomatization

MSC classification

Primary: 03B30: Foundations of classical theories (including reverse mathematics)
Secondary: 83A05: Special relativity 03B10: Classical first-order logic 03B80: Other applications of logic 20G15: Linear algebraic groups over arbitrary fields
Type
Research Article
Information
The Review of Symbolic Logic , Volume 15 , Issue 2 , June 2022 , pp. 334 - 361
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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