Foundations of Physics

, Volume 36, Issue 5, pp 681–714 | Cite as

Twin Paradox and the Logical Foundation of Relativity Theory

  • Judit X. Madarász
  • István Németi
  • Gergely Székely
Article

We study the foundation of space-time theory in the framework of first-order 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 space-time theory (or relativity). First we recall a simple and streamlined FOL-axiomatization 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.

Keywords

twin paradox relativity theory accelerated observers first-order logic axiomatization foundation of relativity theory 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Hilbert D. “Über den Satz von der Gleichheit der Basiswinkel im gleichschenkligen Dreieck”. Proc. London Math. Soc. 35 50 (1902/1903).Google Scholar
  2. 2.
    H. Friedman, On foundational thinking 1, Posting in FOM (Foundations of Mathematics) Archives www.cs.nyu.edu (January 20, 2004).Google Scholar
  3. 3.
    H. Friedman, On foundations of special relativistic kinematics 1, Posting No 206 in FOM (Foundations of Mathematics) Archives www.cs.nyu.edu (January 21, 2004).Google Scholar
  4. 4.
    Väänänen J. (2001). “Second-order logic and foundations of mathematics”. B. Symb. Log. 7:504CrossRefMATHGoogle Scholar
  5. 5.
    H. Andréka, J. X. Madarász, and I. Németi, with contributions from A. Andai, G. Sági, I. Sain, and Cs. Tőke, “On the logical structure of relativity theories,” Research report, Alfréd Rényi Institute of Mathematics, Budapest (2002) http://www.math-inst.hu/pub/algebraic-logic/Contents.html.Google Scholar
  6. 6.
    Ax J. (1978). “The elementary foundations of spacetime”. Found. Phys. 8:507CrossRefADSMathSciNetGoogle Scholar
  7. 7.
    Pambuccian V. (2005). “Axiomatizations of hyperbolic and absolute geometries”. In: Prékopa A. and Molnár E. (eds). Non-Euclidean Geometries. Kluwer, DordrechtGoogle Scholar
  8. 8.
    Andréka H., Madarász J.X., Németi I. (2005). “Logical axiomatizations of space-time”. In: Prékopa A., Molnár E., (eds). Non-Euclidean Geometries. Kluwer, Dordrecht, http://www.math-inst.hu/pub/algebraic-logic/lstsamples.ps.Google Scholar
  9. 9.
    Ferreirós J. (2001). “The road to modern logic – an interpretation”. B. Symb. Log. 7:441CrossRefMATHGoogle Scholar
  10. 10.
    Woleński J. “First-order logic: (philosophical) pro and contra”. In: First-Order Logic Revisited (Logos, Berlin, 2004).Google Scholar
  11. 11.
    Etesi G. and Németi I. (2002). “Non-turing computations via Malament-Hogarth space-times”. Int. J. Theor. Phys. 41:341 arXiv:gr-qc/0104023CrossRefMATHGoogle Scholar
  12. 12.
    Hogarth M. (2004). “Deciding arithmetic using SAD computers”. Brit. J. Phil. Sci. 55:681CrossRefMathSciNetMATHGoogle Scholar
  13. 13.
    Suppes P. (1968). “The desirability of formalization in science”. J. Philos. 65:651CrossRefGoogle Scholar
  14. 14.
    Rudin W. (1953). Principles of Mathematical Analysis. McGraw-Hill, New YorkMATHGoogle Scholar
  15. 15.
    Chang C.C. and Keisler H.J. Model Theory. (North–Holland, Amsterdam, 1973, 1990).Google Scholar
  16. 16.
    d’Inverno R. (1992). Introducing Einstein’s Relativity. Clarendon, OxfordMATHGoogle Scholar
  17. 17.
    Einstein A. (1921). Über die spezielle und die allgemeine Relativitätstheorie. von F. Vieweg, BraunschweigGoogle Scholar
  18. 18.
    Madarász J.X. (2002). Logic and relativity (in the light of definability theory). Eötvös Loránd Univ., Budapest, PhD thesis, http://www.math-inst.hu/pub/algebraic-logic/Contents.html.Google Scholar
  19. 19.
    Székely G. A first order logic investigation of the twin paradox and related subjects. Eötvös Master’s thesis, Loránd University Budapest (2004).Google Scholar
  20. 20.
    Misner C.W., Thorne K.S., Wheeler J.A. (1973). Gravitation. W. H. Freeman, San FranciscoGoogle Scholar
  21. 21.
    Hodges W. (1997). Model Theory. Cambridge University Press, CambridgeMATHGoogle Scholar
  22. 22.
    Tarski A. (1951). A Decision Method for Elementary Algebra and Geometry. University of California, BerkeleyMATHGoogle Scholar
  23. 23.
    Ross K.A. (1980). Elementary Analysis: The Theory of Calculus. Springer, New YorkMATHGoogle Scholar
  24. 24.
    Wald R.M. (1984). General Relativity. Universtiy of Chicago Press, ChicagoMATHGoogle Scholar
  25. 25.
    Taylor E.F. and Wheeler J.A. (2000). Exploring Black Holes: Introduction to General Relativity. Addison Wesley, San FranciscoGoogle Scholar
  26. 26.
    Fuchs L. (1963). Partially Ordered Algebraic Systems. Pergamon, OxfordMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • Judit X. Madarász
    • 1
  • István Németi
    • 1
  • Gergely Székely
    • 1
  1. 1.Alfréd Rényi Institute of Mathematics of the Hungarian Academy of SciencesBudapestHungary

Personalised recommendations