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Gergely Székely

Publications and source records attributed to Gergely Székely.

At least 19 recordsLinked to original sources
arXiv2026

Groups of Worldview Transformations Implied by Einstein's Special Principle of Relativity over Arbitrary Ordered Fields

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.

physics.gen-ph↗
arXiv2025

On Andréka's Conjecture that special relativity is the only possible conceptual reduct of classical kinematics

In this paper, we prove a pure mathematical result which has important implications for the history and philosophy of classical physics and the conceptual origins of relativity theory. In formal terms, we show that, up to definitional equivalence, there is no intermediate model of spacetime lying strictly between special relativity and late classical kinematics. Informally, this means that there was essentially no other option but to switch to special relativity to resolve the conflict between late classical kinematics and the null result of the Michelson--Morley experiment.

math-ph↗
arXiv2025

Definable coordinate geometries over fields, part 1: theory

We define general notions of coordinate geometries over fields and ordered fields, and consider coordinate geometries that are given by finitely many relations that are definable over those fields. We show that the automorphism group of such a geometry determines the geometry up to definitional equivalence; moreover, if we are given two such geometries $\mathcal{G}$ and $\mathcal{G}'$, then the concepts (explicitly definable relations) of $\mathcal{G}$ are concepts of $\mathcal{G}'$ exactly if the automorphisms of $\mathcal{G}'$ are automorphisms of $\mathcal{G}$. We show this by first proving that a relation is a concept of $\mathcal{G}$ exactly if it is closed under the automorphisms of $\mathcal{G}$ and is definable over the field; moreover, it is enough to consider automorphisms that are affine transformations.

math.LO↗
arXiv2025

Definable coordinate geometries over fields, part 2: applications

In Part 1 of this study we showed, for a wide range of geometries, that the relationships between their concept-sets are fully determined by those between their (affine) automorphism groups. In this (self-contained) part, we show how this result can be applied to quickly determine relationships and differences between various geometries and spacetimes, including ordered affine, Euclidean, Galilean, Newtonian, Late Classical, Relativistic and Minkowski spacetimes (we first define these spacetimes and geometries using a Tarskian first-order language centred on the ternary relation $\mathsf{Bw}$ of betweenness). We conclude with a selection of open problems related to the existence of certain intermediate geometries.

math.LO↗
arXiv2025

On variable non-dependence of first-order formulas

In this paper, we introduce a concept of non-dependence of variables in formulas. A formula in first-order logic is non-dependent of a variable if the truth value of this formula does not depend on the value of that variable. This variable non-dependence can be subject to constraints on the value of some variables which appear in the formula, these constraints are expressed by another first-order formula. After investigating its basic properties, we apply this concept to simplify convoluted formulas by bringing out and discarding redundant nested quantifiers. Such convoluted formulas typically appear when one uses a translation function interpreting a theory into another.

math.LO↗
arXiv2021

Complexity in the interdefinability of timelike, lightlike and spacelike relatedness of Minkowski spacetime

Interdefinability of timelike, lightlike and spacelike relatedness of Minkowski spacetime is investigated in detail in the paper, with the aim of finding the simplest definitions. Based on ideas scattered in the literature, definitions are given between any two of these relations that use only 4 variables. All these definitions work over arbitrary Euclidean fields in place of the field of reals, if the dimension n of spacetime is greater than two. If n=2, the definitions work over arbitrary ordered fields except the ones based on lightlike relatedness (where no definition can work by symmetry). None of these relations can be defined from another one using only 3 variables. Our four-variable definitions use only one universal and one existential quantifiers in a specific order. In some of the cases, we show that the order of these quantifiers can be reversed for the price of using twice as many quantifiers. Except two cases, we provide existential/universal definitions using 5 variables or show that no existential/universal definition exists. There are no existential/universal definitions between any two of these relations using only 4 variables. It remains open whether there is an existential (universal) definition of timelike (lightlike) relatedness from spacelike relatedness if $n>2$. Finally, several other open problems related to the quantifier complexity of the simplest possible definitions are given.

math.LO↗
arXiv2021

On the networks of large embeddings

We define a special network that exhibits the large embeddings in any class of similar algebras. With the aid of this network, we introduce a notion of distance that conceivably counts the minimum number of dissimilarities, in a sense, between two given algebras in the class in hand; with the possibility that this distance may take the value $\infty$. We display a number of inspirational examples from different areas of algebra, e.g., group theory and monounary algebras, to show that this research direction can be quite remarkable.

math.GM↗
arXiv2020

Groups of Worldview Transformations Implied by Isotropy of Space

Given any Euclidean ordered field, $Q$, and any 'reasonable' group, $G$, of (1+3)-dimensional spacetime symmetries, we show how to construct a model $M_{G}$ of kinematics for which the set $W$ of worldview transformations between inertial observers satisfies $W=G$. This holds in particular for all relevant subgroups of $Gal$, $cPoi$, and $cEucl$ (the groups of Galilean, Poincaré and Euclidean transformations, respectively, where $c\in Q$ is a model-specific parameter orresponding to the speed of light in the case of Poincaré transformations). In doing so, by an elementary geometrical proof, we demonstrate our main contribution: spatial isotropy is enough to entail that the set $W$ of worldview transformations satisfies either $W\subseteq Gal$, $W\subseteq cPoi$, or $W\subseteq cEucl$ for some $c>0$. So assuming spatial isotropy is enough to prove that there are only 3 possible cases: either the world is classical (the worldview transformations between inertial observers are Galilean transformations); the world is relativistic (the worldview transformations are Poincaré transformations); or the world is Euclidean (which gives a nonstandard kinematical interpretation to Euclidean geometry). This result considerably extends previous results in this field, which assume a priori the (strictly stronger) special principle of relativity, while also restricting the choice of $Q$ to the field of reals. As part of this work, we also prove the rather surprising result that, for any $G$ containing translations and rotations fixing the time-axis $t$, the requirement that $G$ be a subgroup of one of the groups $Gal$, $cPoi$ or $cEucl$ is logically equivalent to the somewhat simpler requirement that, for all $g\in G$: $g[t]$ is a line, and if $g[t]=t$ then $g$ is a trivial transformation (i.e. $g$ is a linear transformation that preserves Euclidean length and fixes the time-axis setwise).

math-ph↗
arXiv2018

Distances between formal theories

In the literature, there have been several methods and definitions for working out if two theories are "equivalent" (essentially the same) or not. In this article, we do something subtler. We provide means to measure distances (and explore connections) between formal theories. We introduce two main notions for such distances. The first one is that of \textit{axiomatic distance}, but we argue that it might be of limited interest. The more interesting and widely applicable notion is that of \textit{conceptual distance} which measures the minimum number of concepts that distinguish two theories. For instance, we use conceptual distance to show that relativistic and classical kinematics are distinguished by one concept only. We also develop further notions of distance, and we include a number of suggestions for applying and extending our project.

math.LO↗
arXiv2018

On Generalization of Definitional Equivalence to Languages with Non-Disjoint Signatures

For simplicity, most of the literature introduces the concept of definitional equivalence only to languages with disjoint signatures. In a recent paper, Barrett and Halvorson introduce a straightforward generalization to languages with non-disjoint signatures and they show that their generalization is not equivalent to intertranslatability in general. In this paper, we show that their generalization is not transitive and hence it is not an equivalence relation. Then we introduce the Andréka and Németi generalization as one of the many equivalent formulations for languages with disjoint signatures. We show that the Andréka-Németi generalization is the smallest equivalence relation containing the Barrett-Halvorson generalization and it is equivalent to intertranslatability even for languages with non-disjoint signatures. Finally, we investigate which definitions for definitional equivalences remain equivalent when we generalize them for theories with non-disjoint signatures.

math.LO↗
arXiv2017

Comparing Classical and Relativistic Kinematics in First-Order Logic

The aim of this paper is to present a new logic-based understanding of the connection between classical kinematics and relativistic kinematics. We show that the axioms of special relativity can be interpreted in the language of classical kinematics. This means that there is a logical translation function from the language of special relativity to the language of classical kinematics which translates the axioms of special relativity into consequences of classical kinematics. We will also show that if we distinguish a class of observers (representing observers stationary with respect to the "Ether") in special relativity and exclude the non-slower-than light observers from classical kinematics by an extra axiom, then the two theories become definitionally equivalent (i.e., they become equivalent theories in the sense as the theory of lattices as algebraic structures is the same as the theory of lattices as partially ordered sets). Furthermore, we show that classical kinematics is definitionally equivalent to classical kinematics with only slower-than-light inertial observers, and hence by transitivity of definitional equivalence that special relativity theory extended with "Ether" is definitionally equivalent to classical kinematics. So within an axiomatic framework of mathematical logic, we explicitly show that the transition from classical kinematics to relativistic kinematics is the knowledge acquisition that there is no "Ether", accompanied by a redefinition of the concepts of time and space.

math-ph↗
arXiv2017

Three Different Formalisations of Einstein's Relativity Principle

We present three natural but distinct formalisations of Einstein's special principle of relativity, and demonstrate the relationships between them. In particular, we prove that they are logically distinct, but that they can be made equivalent by introducing a small number of additional, intuitively acceptable axioms.

physics.class-ph↗
arXiv2016

On some Symmetry Axioms in Relativity Theories

In this paper we review two symmetry axioms of special relativity and their connections to each other together with their role in some famous predictions of relativity theory, such as time dilation, length contraction, and the twin paradox. We also discuss briefly counterparts of these symmetry axioms in general relativity and formulate a conjecture, namely that without them the axioms of general relativity would capture general relativistic spacetimes only up to conformal equivalence.

gr-qc↗
arXiv2014

Why Do the Relativistic Masses and Momenta of Faster-than-Light Particles Decrease as their Speeds Increase?

It has recently been shown within a formal axiomatic framework using a definition of four-momentum based on the Stückelberg-Feynman-Sudarshan-Recami "switching principle" that Einstein's relativistic dynamics is logically consistent with the existence of interacting faster-than-light inertial particles. Our results here show, using only basic natural assumptions on dynamics, that this definition is the only possible way to get a consistent theory of such particles moving within the geometry of Minkowskian spacetime. We present a strictly formal proof from a streamlined axiom system that given any slow or fast inertial particle, all inertial observers agree on the value of $\mathsf{m}\cdot \sqrt{|1-v^2|}$, where $\mathsf{m}$ is the particle's relativistic mass and $v$ its speed. This confirms formally the widely held belief that the relativistic mass and momentum of a positive-mass faster-than-light particle must decrease as its speed increases.

gr-qc↗