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TWIN PARADOX AND THE LOGICAL FOUNDATION OF RELATIVITY THEORY
, 2005
"... Abstract. We study the foundation of spacetime theory in the framework of firstorder logic (FOL). Since the foundation of mathematics has been successfully carried through (via set theory) in FOL, it is not entirely impossible to do the same for spacetime theory (or relativity). First we recall a ..."
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Abstract. We study the foundation of spacetime theory in the framework of firstorder logic (FOL). Since the foundation of mathematics has been successfully carried through (via set theory) in FOL, it is not entirely impossible to do the same for spacetime theory (or relativity). First we recall a simple and streamlined FOLaxiomatization Specrel of special relativity from the literature. Specrel is complete with respect to questions about inertial motion. Then we ask ourselves whether we can prove the usual relativistic properties of accelerated motion (e.g., clocks in acceleration) in Specrel. As it turns out, this is practically equivalent to asking whether Specrel is strong enough to “handle ” (or treat) accelerated observers. We show that there is a mathematical principle called induction (IND) coming from real analysis which needs to be added to Specrel in order to handle situations involving relativistic acceleration. We present an extended version AccRel of Specrel which is strong enough to handle accelerated motion, in particular, accelerated observers. Among others, we show that the Twin Paradox becomes provable in AccRel, but it is not provable without IND. Key words: twin paradox, relativity theory, accelerated observers, firstorder logic, axiomatization, foundation of relativity theory 1.
Logic of spacetime and relativity theory
, 2006
"... 2.1 Motivation for special relativistic kinematics in place of Newtonian kinematics......................... 4 ..."
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2.1 Motivation for special relativistic kinematics in place of Newtonian kinematics......................... 4
Zeno machines and hypercomputation
 Theoretical Computer Science
"... This paper reviews the ChurchTuring Thesis (or rather, theses) with reference to their origin and application and considers some models of “hypercomputation”, concentrating on perhaps the most straightforward option: Zeno machines (Turing machines with accelerating clock). The halting problem is br ..."
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This paper reviews the ChurchTuring Thesis (or rather, theses) with reference to their origin and application and considers some models of “hypercomputation”, concentrating on perhaps the most straightforward option: Zeno machines (Turing machines with accelerating clock). The halting problem is briefly discussed in a general context and the suggestion that it is an inevitable companion of any reasonable computational model is emphasised. It is suggested that claims to have “broken the Turing barrier ” could be toned down and that the important and wellfounded rôle of Turing computability in the mathematical sciences stands unchallenged.
Definability as hypercomputational effect
 Applied Mathematics and Computation
"... The classical simulation of physical processes using standard models of computation is fraught with problems. On the other hand, attempts at modelling realworld computation with the aim of isolating its hypercomputational content have struggled to convince. We argue that a better basic understandin ..."
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The classical simulation of physical processes using standard models of computation is fraught with problems. On the other hand, attempts at modelling realworld computation with the aim of isolating its hypercomputational content have struggled to convince. We argue that a better basic understanding can be achieved through computability theoretic deconstruction of those physical phenomena most resistant to classical simulation. From this we may be able to better assess whether the hypercomputational enterprise is proleptic computer science, or of mainly philosophical interest.
The extent of computation in MalamentHogarth spacetimes
, 2008
"... We analyse the extent of possible computations following Hogarth [7] in MalamentHogarth (MH) spacetimes, and Etesi and Németi [3] in the special subclass containing rotating Kerr black holes. [7] had shown that any arithmetic statement could be resolved in a suitable MH spacetime. [3] had shown tha ..."
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We analyse the extent of possible computations following Hogarth [7] in MalamentHogarth (MH) spacetimes, and Etesi and Németi [3] in the special subclass containing rotating Kerr black holes. [7] had shown that any arithmetic statement could be resolved in a suitable MH spacetime. [3] had shown that some ∀ ∃ relations on natural numbers which are neither universal nor couniversal, can be decided in Kerr spacetimes, and had asked specifically as to the extent of computational limits there. The purpose of this note is to address this question, and further show that MH spacetimes can compute far beyond the arithmetic: effectively Borel statements (so hyperarithmetic in second order number theory, or the structure of analysis) can likewise be resolved: Theorem A. If H is any hyperarithmetic predicate on integers, then there is an MH spacetime in which any query?n ∈ H? can be computed. In one sense this is best possible, as there is an upper bound to computational ability in any spacetime which is thus a universal constant of the spacetime M. Theorem C. Assuming the (modest and standard) requirement that spacetime manifolds be paracompact and Hausdorff, for any MH spacetime M there will be a countable ordinal upper bound, w(M), on the complexity of questions in the Borel hierarchy resolvable in it.
Abstract geometrical computation: Turingcomputing ability and undecidability
, 2004
"... In the Cellular Automata (CA) literature, discrete lines inside (discrete) spacetime diagrams are often idealized as Euclidean lines in order to analyze a dynamics or to design CA for special purposes. In this article, we present a parallel analog model of computation corresponding to this ideali ..."
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In the Cellular Automata (CA) literature, discrete lines inside (discrete) spacetime diagrams are often idealized as Euclidean lines in order to analyze a dynamics or to design CA for special purposes. In this article, we present a parallel analog model of computation corresponding to this idealization: dimensionless signals are moving on a continuous space in continuous time generating Euclidean lines on (continuous) spacetime diagrams. Like CA, this model is parallel, synchronous, uniform in space and time, and uses local updating. The main difference is that space and time are continuous and not discrete (i.e. R instead of Z). In this article, the model is restricted to Q in order to remain inside Turingcomputation theory. We prove that our model can carry out any Turingcomputation through twocounter automata simulation and provide some undecidability results.
Characteristics of discrete transfinite time Turing machine models: halting times, stabilization times, and . . .
, 2008
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Visualizing some ideas about Gödeltype rotating universes
, 2008
"... Some kinds of physical theories describe what our universe looks like. Other kinds of physical theories describe instead what the universe could be like independently of the properties of the actual universe. This second kind aims for the “basic laws of physics” in some sense which we will not make ..."
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Some kinds of physical theories describe what our universe looks like. Other kinds of physical theories describe instead what the universe could be like independently of the properties of the actual universe. This second kind aims for the “basic laws of physics” in some sense which we will not make precise here (but cf. e.g. Malament [25, pp.9899]). The present paper belongs to the second kind. Moreover, it is even more abstract than this, namely it aims for visualizing or grasping some mathematical or logical aspects of what the universe could be like. The first six pages of this material are of a “sciencepopularizing ” character in the sense that first we recall a spacetime diagram from HawkingEllis [18] as “Godgiven truth”, i.e. we do not explain why the reader should believe that diagram. Then we derive carefully in an easily understandable visual manner an exciting, exotic consequence of that diagram: timetravel. This applies to the first six pages. The rest of this work is of a more ambitious character. The reader does not have to believe anything 1. We do our best to make the paper selfcontained and explain and visualize most of what we say. In more detail, this work consists of Sections 18. Section 1 (p.2) is the just mentioned “popular ” part. Section 2 (p.8) lays the foundation for discussing rotating universes. E.g. it shows how to visualize such spacetimes. The spacetime built up in this section is called the “Naive Spiral world”. Section 3 (p.19) is about nonexistence of a natural “now ” in Gödel’s universe GU. Section 4 (p.22) introduces corotating coordinates “transforming the rotation away”. Section 5 (p.29) refines the Gödeltype universe (obtained in Section 2). Section 6 (p.46) illustrates a fuller view of the refined version of GU. Section 7 (p.52) recoordinatizes the refined GU in order that the socalled gyroscopes do not rotate in this coordinatization. Section 8 (p.67) gives connections with the literature. E.g. it presents detailed computational comparison with the spacetime metric in Gödel’s papers. Section 9 (p.70) contains technical data about how we constructed the figures illustrating Gödel’s universe. 1 Not even the diagram recalled from HawkingEllis [18] in Figure 1 or any of the statements made in the first six pages.