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A Logic Based Foundation and Analysis of Relativity Theory and LogicPage: 1

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Special Relativity Relativity Theory and LogicPage: 2

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Relativity Theory and LogicPage: 3 Bodies (test particles), Inertial Observers, Photons, Quantities, usual operations on it, Worldview B IOb Ph 0 Q = number-line W

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Relativity Theory and LogicPage: 4 W(m, t x y z, b) body “b” is present at coordinates “t x y z” for observer “m” t x y b (worldline) m

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1. ( Q, +, ∙ ) is a field in the sense of abstract algebra (with 0, −, 1, / as derived operations) 2. 3. Ordering derived: Relativity Theory and LogicPage: 5 AxField Usual properties of addition and multiplication on Q : Q is an ordered Euclidean field.

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Relativity Theory and LogicPage: 6 AxPh For all inertial observers the speed of light is the same in all directions and is finite. In any direction it is possible to send out a photon. t x y ph 1 ph 2 ph 3 Formalization:

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Relativity Theory and LogicPage: 7 AxEv All inertial observers coordinatize the same events. Formalization: t x y WmWm b1b1 b2b2 m

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Relativity Theory and LogicPage: 8 Formalization: AxSelf An inertial observer sees himself as standing still at the origin. t x y t = worldline of the observer m WmWm

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Relativity Theory and LogicPage: 9 AxSymd Any two observers agree on the spatial distance between two events, if these two events are simultaneous for both of them, and |v m (ph)|=1. Formalization: k t x y m x y t e1 e2 ph is the event occurring at p in m’s worldview

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v m (b) Relativity Theory and LogicPage: 10 What is speed? m b p ptpt psps qsqs qtqt q 1

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Relativity Theory and LogicPage: 11 SpecRel = {AxField, AxPh, AxEv, AxSelf, AxSymd} SpecRel Theorems Proofs …

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Relativity Theory and LogicPage: 12 Thm2 Thm1

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t x y m Relativity Theory and LogicPage: 13 Proof of Thm2 (NoFTL): e1 k ph - AxSelf - AxPh e2 ph1 e3 e2 ph1 - AxEv t x y k e1 - AxField ph p q Tangent plane k asks m : where did ph and ph1 meet?

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Conceptual analysis Which axioms are needed and why Project: Find out the limits of NoFTL How can we weaken the axioms to make NoFTL go away Relativity Theory and LogicPage: 14

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Igor D. Novikov, September 3 Budapest: Relativity Theory and LogicPage: 15 It is an important research today to study spacetimes with more than one time dimensions

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Thm3 SpecRel is consistent Relativity Theory and LogicPage: 16 Thm4 No axioms of SpecRel is provable from the rest Thm5 SpecRel is complete with respect to Minkowski geometries (e.g. implies all the basic paradigmatic effects of Special Relativity - even quantitatively!)

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Relativity Theory and LogicPage: 17 Thm6 SpecRel generates an undecidable theory. Moreover, it enjoys both of Gödel’s incompleteness properties Thm7 SpecRel has a decidable extension, and it also has a hereditarily undecidable extension. Both extensions are physically natural.

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Relativity Theory and LogicPage: 18 Moving clocks get out of synchronism. Captain claims that the two clocks show the same time. Thm8 v k (m)

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Relativity Theory and LogicPage: 19 Thm8 (formalization of clock asynchronism) (1) Assume e, e’ are separated in the direction of motion of m in k’s worldview, m ykyk k xkxk 1t1t v 1x1x xmxm e e’ |v| Assume SpecRel. Assume m,k IOb and events e, e’ are simultaneous for m, m xmxm e’ e

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Relativity Theory and LogicPage: 20 (2) e, e’ are simultaneous for k, too e, e’ are separated orthogonally to v k (m) in k’s worldview m xmxm ykyk k xkxk ymym e e’

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Relativity Theory and LogicPage: 21 Thought-experiment for proving relativity of simultaneity.

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Relativity Theory and LogicPage: 22 Thought-experiment for proving relativity of simultaneity.

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Relativity Theory and LogicPage: 23

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The Vienna Circle and Hungary, Vienna, May 19-20, 2008Relativity Theory and LogicPage: 24

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Relativity Theory and LogicPage: 25 Moving clocks tick slowly Moving spaceships shrink Thm9

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Relativity Theory and LogicPage: 26 Thm9 (formalization of time-dilation) Assume SpecRel. Let m,k IOb and events e, e’ are on k’s lifeline. m xmxm e e’ k 1 v k xkxk e

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Relativity Theory and LogicPage: 27

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Relativity Theory and LogicPage: 28 Thm10 (formalization of spaceship shrinking) Assume SpecRel. Let m,k,k’ IOb and assume m xmxm k k’ e’ e k x k’

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Relativity Theory and LogicPage: 29 Experiment for measuring distance (for m) by radar: m’ m’’ m e e’ Dist m (e,e’)

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Relativity Theory and LogicPage: 30

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Relativity Theory and LogicPage: 31 v = speed of spaceship

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Relativity Theory and LogicPage: 32 t x y WmWm b1b1 b2b2 m w mk ev m ev k p q

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Relativity Theory and LogicPage: 33 Thm11 The worldview transformations w mk are Lorentz transformations (composed perhaps with a translation).

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Relativity Theory and LogicPage: 34 Minkowskian Geometry: Top-down approach to SR, observer-free. Robert Goldblatt: complete FOL axiom system MinkGeo

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Relativity Theory and LogicPage: 35 Interpretations Definitional equivalence

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Relativity Theory and LogicPage: 36 Minkowskian Geometry James Ax: Signals Alfred Robb: causality Patrick Suppes: worldview transformations … Connections between theories. Dynamics of theories. Interpretations between them. Theory morphisms. Definitional equivalence. Tamás Füzessy, Judit Madarász and Gergely Székely began joint work in this Definability theory of logic! (Tarski, Makkai) Contribution of relativity to logic: definability theory with new objects definable (and not only with new relations definable). J. Madarász’ dissertation.

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Relativity Theory and LogicPage: 37 SpecRel = {AxField, AxPh, AxEv, AxSelf, AxSymd} SpecRel Theorems Proofs …

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Conceptual analysis of SR goes on … on our homepage Relativity Theory and LogicPage: 38 New theory is coming:

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