Abstract
This paper is an amalgam of physics and mathematical logic. It contains an elementary axiomatization of spacetime in terms of the primitive concepts of particle, signal, and transmission and reception. In the elementary language formed with these predicates we state AxiomsE, C, andU, which are naturally interpretable as basic physical properties of particles and signals. We then determine all mathematical models of this axiom system; these represent certain generalizations of the standard model. Also, the automorphism groups of the models are determined. Finally we give another physical model and discuss the philosophical implications.
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References
H. Weyl,Philosophy of Mathematics and the Natural Sciences (Princeton Univ. Press, Princeton, 1949).
J. Graves,The Conceptual Foundation of Contemporary Relativity Theory (MIT Press, Cambridge, Mass., 1971).
T. Gold,The Nature of Time (Cornell Univ. Press, Ithaca, New York, 1967).
K. Ford,Basic Physics (Blaisdell, Waltham, Mass., 1968).
I. Segal,Mathematical Cosmology and Extragalactic Astronomy (Academic Press, New York, 1976).
L. Brillouin,Relativity Reexamined (Academic Press, New York, 1970).
H.-Y. Chiu and W. Hoffman,Gravity and Relativity (Benjamin, New York, 1964).
N. Rosen,Astrophys. J. 211, 357 (1977).
R. Penrose, inBattelle Rencontres 1967 (Benjamin, New York, 1968), esp. pp. 211ff.
J. Wheeler, inBattelle Rencontres 1967 (Benjamin, New York, 1968), esp. pp. 242ff.
I. Segal,Mathematical Problems in Relativistic Physics (AMS, Providence, 1963), esp. Ch. VIII.
P. Kelemen and A. Robinson,J. Math. Phys. 13, 1870 (1972).
E. Wigner, Quantum mechanical distribution functions revisited, inPerspectives in Quantum Theory (MIT Press, 1971), pp. 25–36.
J. Ax,Found. Phys. 6, 371 (1976).
A. Einstein,Ideas and Opinions (Laurel Edition, Crown, New York, 1973).
A. Einstein,Relativity (Crown, New York, 1961).
H. Reichenbach,Axiomatization of the Theory of Relativity (Univ. of Calif. Press, Berkeley, 1969).
S. Basri,A Deductive Theory of Space and Time (North-Holland, Amsterdam, 1966).
R. Marzke and J. Wheeler, Gravity as Geometry, inGravitation and Relativity, Chiu and Hoffman, eds. (Benjamin, New York, 1964).
J. Schutz,Foundations of Special Relativity: Kinematic Axioms for Minkowski Space-Time (Springer Verlag, New York, 1973).
A. A. Robb,A Theory of Time and Space (Cambridge, 1914).
A. D. Alexandrov,Can. J. Math. 19, 1119 (1967).
A. D. Alexandrov and V. V. Ovchinnikova,Vestnik Leningr. Univ. 11, 95 (1953).
E. Zeeman,J. Math. Phys. 5, 490 (1964).
P. Cohen,Set Theory and the Continuum Hypothesis (Benjamin, New York, 1966); A. Robinson,Introduction to Model Theory (North-Holland, Amsterdam, 1965).
E. Mendelson,Introduction to Mathematical Logic (Van Nostrand, Princeton, 1968).
A. Tarski,The Axiomatic Method with Special Reference to Geometry and Physics (North-Holland, Amsterdam, 1959).
A. Tarski,A Decision Method for Elementary Algebra and Geometry, 2nd ed. (Berkeley and Los Angeles, 1951).
D. Hilbert,The Foundations of Geometry (transl. by E. V. Townsend), 3rd ed. (Open Court, LaSalle, Ill., 1938).
L. Blumenthal,A Modern View of Geometry (Freeman, San Francisco, 1961).
E. Artin,Geometric Algebra (Interscience, New York, 1957).
H. A. Lorentz, A. Einstein, H. Minkowski, and H. Weyl,The Principle of Relativity (Dover, New York, 1952).
R. Feynman, R. Leighton, and M. Sands,Lectures on Physics, Vol. I (Addison-Wesley, New York, 1966).
B. Russel,A History of Western Philosophy (Simon and Schuster, New York, 1945).
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Supported by National Science Foundation under Grant No. MP575-09371.
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Ax, J. The elementary foundations of spacetime. Found Phys 8, 507–546 (1978). https://doi.org/10.1007/BF00717578
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DOI: https://doi.org/10.1007/BF00717578