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Logical axiomatizations of spacetime. Samples from the literature
 In: NonEuclidean Geometries (J'anos Bolyai Memorial Volume
, 2005
"... Abstract We study relativity theory as a theory in the sense of mathematical logic. We use firstorder logic (FOL) as a framework to do so. We aim at an “analysis of the logical structure of relativity theories”. First we build up (the kinematics of) special relativity in FOL, then analyze it, and t ..."
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Abstract We study relativity theory as a theory in the sense of mathematical logic. We use firstorder logic (FOL) as a framework to do so. We aim at an “analysis of the logical structure of relativity theories”. First we build up (the kinematics of) special relativity in FOL, then analyze it, and then we experiment with generalizations in the direction of general relativity. The present paper gives samples from an ongoing broader research project which in turn is part of a research direction going back to Reichenbach and others in the 1920’s. We also try to give some perspective on the literature related in a broader sense. In the perspective of the present work, axiomatization is not a final goal. Axiomatization is only a first step, a tool. The goal is something like a conceptual analysis of relativity in the framework of logic. In section 1 we recall a complete FOLaxiomatization Specrel of special relativity from [5],[31]. In section 2 we answer questions from papers by Ax and Mundy concerning the logical status of faster than light motion (FTL) in relativity. We claim that already very small/weak fragments of Specrel prove “No FTL”. In section 3 we give a sketchy outlook for the possibility of generalizing Specrel to theories permitting accelerated observers (gravity). In section 4 we continue generalizing Specrel in the direction of general relativity by localizing it, i.e. by replacing it with a version still in firstorder logic but now local (in the sense of general relativity theory). In section 5 we give samples from the broader literature.
Firstorder logic foundation of relativity theories
 In New Logics for the XXIst Century II, Mathematical Problems from Applied Logics, volume 5 of International Mathematical Series
, 2006
"... Abstract. 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 ..."
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Cited by 8 (8 self)
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Abstract. 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.
Twin Paradox and the logical foundation of spacetime. Foundation of Physics
"... 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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Cited by 7 (6 self)
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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 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. 1.
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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Cited by 7 (6 self)
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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.
SecondOrder Description Logics: Semantics, Motivation, and a Calculus
"... paper [6] describing the evolution of the current web from a humanprocessable environment to a machineprocessable one. The basic idea was to annotate web resources and give them a machineprocessable meaning; the Semantic Web was born. ..."
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paper [6] describing the evolution of the current web from a humanprocessable environment to a machineprocessable one. The basic idea was to annotate web resources and give them a machineprocessable meaning; the Semantic Web was born.
AXIOMATIZING RELATIVISTIC DYNAMICS WITHOUT CONSERVATION POSTULATES
"... Abstract. A part of relativistic dynamics is axiomatized by simple and purely geometrical axioms formulated within firstorder logic. A geometrical proof of the formula connecting relativistic and rest masses of bodies is presented, leading up to a geometric explanation of Einstein’s famous E = mc 2 ..."
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Cited by 3 (3 self)
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Abstract. A part of relativistic dynamics is axiomatized by simple and purely geometrical axioms formulated within firstorder logic. A geometrical proof of the formula connecting relativistic and rest masses of bodies is presented, leading up to a geometric explanation of Einstein’s famous E = mc 2. The connection of our geometrical axioms and the usual axioms on the conservation of mass, momentum and fourmomentum is also investigated. 1.
Is the Continuum Hypothesis a definite mathematical problem?
"... [t]he analysis of the phrase “how many ” unambiguously leads to a definite meaning for the question [“How many different sets of integers do their exist?”]: the problem is to find out which one of the א’s is the number of points of a straight line … Cantor, after having proved that this number is gr ..."
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[t]he analysis of the phrase “how many ” unambiguously leads to a definite meaning for the question [“How many different sets of integers do their exist?”]: the problem is to find out which one of the א’s is the number of points of a straight line … Cantor, after having proved that this number is greater than א0, conjectured that it is א1. An equivalent proposition is this: any infinite subset of the continuum has the power either of the set of integers or of the whole continuum. This is Cantor’s continuum hypothesis. … But, although Cantor’s set theory has now had a development of more than sixty years and the [continuum] problem is evidently of great importance for it, nothing has been proved so far relative to the question of what the power of the continuum is or whether its subsets satisfy the condition just stated, except that … it is true for a certain infinitesimal fraction of these subsets, [namely] the analytic sets. Not even an upper bound, however high, can be assigned for the power of the continuum. It is undecided whether this number is regular or singular, accessible or inaccessible, and (except for König’s negative result) what its character of cofinality is. Gödel 1947, 516517 [in Gödel 1990, 178]
A logical analysis of the timewarp effect of general relativity,” in preparation
"... Abstract. Several versions of the Gravitational Time Dilation effect of General Relativity are formulated by the use of Einstein’s Equivalence Principle. It is shown that all of them are logical consequence of a firstorder axiom system of Special Relativity extended to accelerated observers. 1. ..."
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Abstract. Several versions of the Gravitational Time Dilation effect of General Relativity are formulated by the use of Einstein’s Equivalence Principle. It is shown that all of them are logical consequence of a firstorder axiom system of Special Relativity extended to accelerated observers. 1.
From IF to BI A Tale of Dependence and Separation
"... Abstract. We take a fresh look at the logics of informational dependence and independence of Hintikka and Sandu and Väänänen, and their compositional semantics due to Hodges. We show how Hodges ’ semantics can be seen as a special case of a general construction, which provides a context for a useful ..."
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Abstract. We take a fresh look at the logics of informational dependence and independence of Hintikka and Sandu and Väänänen, and their compositional semantics due to Hodges. We show how Hodges ’ semantics can be seen as a special case of a general construction, which provides a context for a useful completeness theorem with respect to a wider class of models. We shed some new light on each aspect of the logic. We show that the natural propositional logic carried by the semantics is the logic of Bunched Implications due to Pym and O’Hearn, which combines intuitionistic and multiplicative connectives. This introduces several new connectives not previously considered in logics of informational dependence, but which we show play a very natural rôle, most notably intuitionistic implication. As regards the quantifiers, we show that their interpretation in the Hodges semantics is forced, in that they are the image under the general construction of the usual Tarski semantics; this implies that they are adjoints to substitution, and hence uniquely determined. As for the dependence predicate, we show that this is definable from a simpler predicate, of constancy or dependence on nothing. This makes essential use of the intuitionistic implication. The Armstrong axioms for functional dependence are then recovered as a standard set of axioms for intuitionistic implication. We also prove a full abstraction result in the style of Hodges, in which the intuitionistic implication plays a very natural rôle. 1.
NonStandard Models of Arithmetic: a Philosophical and Historical perspective MSc Thesis (Afstudeerscriptie)
, 2010
"... 1 Descriptive use of logic and Intended models 1 1.1 Standard models of arithmetic.......................... 1 1.2 Axiomatics and Formal theories......................... 3 1.3 Hintikka and the two uses of logic in mathematics.............. 5 ..."
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1 Descriptive use of logic and Intended models 1 1.1 Standard models of arithmetic.......................... 1 1.2 Axiomatics and Formal theories......................... 3 1.3 Hintikka and the two uses of logic in mathematics.............. 5