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We study the foundation of space-time theory in the framework of first-orderlogic (FOL). Since the foundation of mathematics has been successfully carriedthrough (via set theory) in FOL, it is not entirely impossible to do the samefor space-time theory (or relativity). First we recall a simple and streamlinedFOL-axiomatization SpecRel of special relativity from the literature. SpecRelis complete with respect to questions about inertial motion. Then we askourselves whether we can prove usual relativistic properties of acceleratedmotion (e.g., clocks in acceleration) in SpecRel. As it turns out, this ispractically equivalent to asking whether SpecRel is strong enough to "handle"(or treat) accelerated observers. We show that there is a mathematicalprinciple called induction (IND) coming from real analysis which needs to beadded to SpecRel in order to handle situations involving relativisticacceleration. We present an extended version AccRel of SpecRel which is strongenough to handle accelerated motion, in particular, accelerated observers.Among others, we show that the Twin Paradox becomes provable in AccRel, but itis not provable without IND.Comment: 24 pages, 6 figure

Twin Paradox and the Logical Foundation of Relativity Theory