2 Special relativity 4 2.1 Motivation for special relativistic kinematics in place of Newtonian kinematics . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.3 Axiomatization Specrel of special relativity in first-order logic 15 2.4 Characteristic differences between Newtonian and special relativistic kinematics . . . . . . . . . . . . . . . . . . . . . . . . 22 2.5 Explicit description of all models of Specrel, basic logical investigations . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.6 Observer-independent geometries in relativity theory; duality and definability theory of logic . . . . . . . . . . . . . . . . . 47 2.7 Conceptual analysis and “reverse relativity” . . . . . . . . . . 59

On generalizing the logic-approach to space-time towards general relativity: first steps

Logic of Space-Time and Relativity Theory

Applying mathematical logic in foundations of relativity theories is not a new idea at all, it goes back to such leading mathematicians and philosophers as Hilbert, Reichenbach, Carnap, Godel, Tarski, Suppes and Friedman among others. The work of our school of Logic and Relativity led by Andreka and Nemeti is continuation to their research.

This thesis is mainly about extensions of the first-order logic axiomatization of special relativity introduced by Andreka, Madarasz and Nemeti. These extensions include extension to accelerated observers, relativistic dynamics and general relativity; however, its main subject is the extension to accelerated observers (AccRel). One surprising result is that natural extension to accelerated observers is not enough if we want our theory to imply certain experimental facts, such as the twin paradox. Even if we add the whole first-order theory of real numbers to this natural extension, it is still not enough to imply the twin paradox. Nevertheless, that does not mean that this task cannot be carried out within first-order logic since by approximating a second-order logic axiom of real numbers, we introduce a first-order axiom schema that solves the problem. Our theory AccRel nicely fills the gap between special and general relativity theories, and only one natural generalization step is needed to achieve a first-order logic axiomatization of general relativity from it. We also show that AccRel is strong enough to make predictions about the gravitational effect slowing down time.

Our general aims are to axiomatize relativity theories within pure first-order logic using simple, comprehensible and transparent basic assumptions (axioms); to prove the surprising predictions (theorems) of relativity theories from a few convincing axioms; to eliminate tacit assumptions from relativity by replacing them with explicit axioms formulated in first-order logic (in the spirit of the first-order logic foundation of mathematics and Tarski's axiomatization of geometry); and to investigate the relationship between the axioms and the theorems.

/pdf/first-order-logic-investigation-of-relativity-theory-with-an-9o3nebb4pm.pdf

First-Order Logic Investigation of Relativity Theory with an Emphasis on Accelerated Observers

This thesis is mainly about extensions of the first-order logic axiomatization of special relativity introduced by Andr\'eka, Madar\'asz and N\'emeti. These extensions include extension to accelerated observers, relativistic dynamics and general relativity; however, its main subject is the extension to accelerated observers (AccRel). One surprising result is that natural extension to accelerated observers is not enough if we want our theory to imply certain experimental facts, such as the twin paradox. Even if we add the whole first-order theory of real numbers to this natural extension, it is still not enough to imply the twin paradox. Nevertheless, that does not mean that this task cannot be carried out within first-order logic since by approximating a second-order logic axiom of real numbers, we introduce a first-order axiom schema that solves the problem. Our theory AccRel nicely fills the gap between special and general relativity theories, and only one natural generalization step is needed to achieve a first-order logic axiomatization of general relativity from it. We also show that AccRel is strong enough to make predictions about the gravitational effect slowing down time. Our general aims are to axiomatize relativity theories within pure first-order logic using simple, comprehensible and transparent basic assumptions (axioms); to prove the surprising predictions (theorems) of relativity theories from a few convincing axioms; to eliminate tacit assumptions from relativity by replacing them with explicit axioms formulated in first-order logic (in the spirit of the first-order logic foundation of mathematics and Tarski's axiomatization of geometry); and to investigate the relationship between the axioms and the theorems.

Motivation and perspective for an exciting new research direction interconnecting logic, spacetime theory, relativity—including such revolutionary areas as black hole physics, relativistic computers, new cosmology—are presented in this paper. We would like to invite the logician reader to take part in this grand enterprise of the new century. Besides general perspective and motivation, we present initial results in this direction.

/pdf/first-order-logic-foundation-of-relativity-theories-4kx2sjsy8g.pdf

First-Order Logic Foundation of Relativity Theories

Within an axiomatic framework of kinematics, we prove that the existence of faster than light particles is logically independent of Einstein's special theory of relativity. Consequently, it is consistent with the kinematics of special relativity that there might be faster than light particles.

On generalizing the logic-approach to space-time towards general relativity: first steps

Citations

Logic of Space-Time and Relativity Theory

First-Order Logic Investigation of Relativity Theory with an Emphasis on Accelerated Observers

First-Order Logic Investigation of Relativity Theory with an Emphasis on Accelerated Observers

First-Order Logic Foundation of Relativity Theories

The existence of superluminal particles is consistent with the kinematics of Einstein's special theory of relativity

Related Papers (5)

The elementary foundations of spacetime

Model Theory

The Desirability of Formalization in Science

Logic of Space-Time and Relativity Theory

Relativistic computers and the Turing barrier