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Andreka, H., Madarasz, J. X. and Nemeti, I.

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Logical axiomatizations of space-time. Samples from the literature

Abstract: We study relativity theory as a theory in the sense of mathematical
logic. We use first-order 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 FOL-axiomatization Specrel of
special relativity from [1],
[2]. 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 observer''. 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 first-order logic but now local (in the sense of general relativity
theory). In section 5 we give samples from the broader literature.

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