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Set theory and physics

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Abstract

Inasmuch as physical theories are formalizable, set theory provides a framework for theoretical physics. Four speculations about the relevance of set theoretical modeling for physics are presented: the role of transcendental set theory (i) in chaos theory, (ii) for paradoxical decompositions of solid three-dimensional objects, (iii) in the theory of effective computability (Church-Turing thesis) related to the possible “solution of supertasks,” and (iv) for weak solutions. Several approaches to set theory and their advantages and disadvatages for physical applications are discussed: Canlorian “naive” (i.e., nonaxiomatic) set theory, contructivism, and operationalism. In the author's opinion, an attitude of “suspended attention” (a term borrowed from psychoanalysis) seems most promising for progress. Physical and set theoretical entities must be operationalized wherever possible. At the same time, physicists should be open to “bizarre” or “mindboggling” new formalisms, which need not be operationalizable or testable at the lime of their creation, but which may successfully lead to novel fields of phenomenology and technology.

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References

  1. A. Einstein,Sitzungberichte der PreuΒischen Akademie der Wissenschaften 1, 123 (1921); reprinted in A. Einstein,Mein Weltbild (Ullstein, 1988), pp. 119–120.

    Google Scholar 

  2. G. Cantor, “BeitrÄge zur Begründung der transfiniten Mengenlehre,”Math. Annal. 46, 481–512 (1895);ibid. 49, 207–246 (1897); reprinted in Ref. 4.

    Google Scholar 

  3. D. Hilbert, “über das Unendliche,”Math. Ann. 95, 161–190 (1926).

    Google Scholar 

  4. G. Cantor,Gesammelte Abhandlungen, A. Fraenkel and E. Zermelo, eds. (Springer, Berlin, 1932).

    Google Scholar 

  5. W. Boos, “Consistency and Konsistenz,”Erkenntnis 26, 1–43 (1987).

    Google Scholar 

  6. A. A. Fraenkel, Y. Bar-Hillel, and A. Levy,Foundations of Set Theory, 2nd revised edition (North Holland, Amsterdam, 1984).

    Google Scholar 

  7. H. Wang,Reflections on Kurt Gödel (MIT Press, Cambridge, Massachusetts, 1991).

    Google Scholar 

  8. K. Gödel,Monatsh. Math. Phys. 38, 173 (1931).

    Google Scholar 

  9. D. Bridges and F. Richman,Varieties of Constructive Mathematics (Cambridge University Press, Cambridge, 1987).

    Google Scholar 

  10. E. Bishop and D. S. Bridges,Conservative Analysis (Springer, Berlin, 1985).

    Google Scholar 

  11. P. W. Bridgman, “A physicist's second reaction to Mengenlehre,”Scr. Math. 2, 101–117; 224–234(1934).

    Google Scholar 

  12. R. Landauer, “Wanted: A physically possible theory of physics,” inIEEE Spectrum 4, 105–109 (1967).

    Google Scholar 

  13. R. Landauer, “Fundamental physical limitations of the computional process; an informal commentary,” inCybernetics Machine Group Newssheet 1/1/87.

  14. R. Landauer, “Advertisement for a paper I like,” inOn Limits, J. L. Casti and J. F. Traub, eds. (Santa Fe Institute Report 94-10-056, Santa Fe, New Mexico, 1994), p. 39.

  15. P. W. Bridgman,The Logic of Modern Physics (Macmillan, New York, 1927).

    Google Scholar 

  16. P. W. Bridgman,The Nature of Physical Theory (Princeton University Press, Princeton, 1936).

    Google Scholar 

  17. P. W. Bridgman,Reflections of a Physicist (Philosophical Library, New York, 1950).

    Google Scholar 

  18. P. W. Bridgman,The Nature of Some of Our Physical Cocepts (Philosophical Library, New York, 1952).

    Google Scholar 

  19. R. O. Gandy, “Limitations to mathematical knowledge,” inLogic Colloquium '82, D. van Dalen, D. Lascar, and J. Smiley, eds. (North-Holland, Amsterdam, 1982), pp. 129–146.

    Google Scholar 

  20. R. O. Gandy, “Church's thesis and principles for mechanics,” inThe Kleene Symposium, J. Barwise, H. J. Kreisler and K. Kunen, eds. (North-Holland), Amsterdam, 1980), pp. 123–148.

    Google Scholar 

  21. D. Mundici, “Irreversibility, uncerainty, relativity and computer limitations,”Nuovo Cimento 61, 297–305 (1981).

    Google Scholar 

  22. R. Landauer, “John Casti's page on “finiteness and real-world limits”, inOn Limits, J. L. Casti and J. F. Traub, eds. (Santa Fe Institute Report 94-10-056, Santa Fe, New Mexico, 1994), p. 34.

  23. J. L. Casti, “Finiteness and real-world limits,” inOn Limits, J. L. Casti and J. F. Traub eds., (Santa Fe Institute Report 94-10-056, Santa Fe, New Mexico, 1994), p. 34.

  24. E. Specker,Selecta (BirkhÄuser Verlag, Basel, 1990).

    Google Scholar 

  25. P. S. Wang, “The undecidability of the existence of zeros of real elementary functions,”J. Assoc. Comput. Mach. 21, 586–589 (1974).

    Google Scholar 

  26. G. Kreisel, “A notion of mechanistic theory,”Synthese 29, 11–26 (1974).

    Google Scholar 

  27. D. Stefanescu,Mathematical Models in Physics (University of Bucharest Press, Bucharest, 1984) (in Romanian).

    Google Scholar 

  28. M. Pour-El, I. Richards,Computability in Analysis and Physics (Springer-Verlag, Berlin, 1989).

    Google Scholar 

  29. R. Penrose.The Emperor's New Mind. Concerning Computers, Minds, and the Laws of Physics (Vintage, London, 1990). (First published by Oxford University Press, Oxford, 1989).

    Google Scholar 

  30. D. S. Bridges, “Constructive mathematics and unbounded operators—a reply to Hellman,”J. Philos. Logic. in press.

  31. C. Calude, D. I. Cambell, K. Svozil, and D. Stefanescu, “Strong determinism vs. computability,”e-print quant-ph/9412004.

  32. G. J. Chaitin,Information, Randomness and Incompleteness, 2nd edn. (World Scientific, Signapore, 1987, 1990);Algorithmic Information Theory (Cambridge University Press, Cambridge, 1987):Information-Theoretic Incompleteness (World Scientific, Singapore, 1992).

    Google Scholar 

  33. C. Calude,Information and Randomness—An Algorithmic Perspectiv (Springer, Berlin, 1994).

    Google Scholar 

  34. M. Li and P. M. B. Vitányi, “Kolmogorov complexity and its applications,” inHandbook of Theoretical Computer Sciences, Algorithmcs and Complexity, Volume A (Elsevier, Amsterdam, and MIT Press, Cambridge, Massachusetts, 1990).

    Google Scholar 

  35. R. Shaw,Z. Naturforsch. 36a, 80 (1981).

    Google Scholar 

  36. St. Smale,The Mathematics of Time (Springer, New York, 1980).

    Google Scholar 

  37. J. Ford,Phys. Today 40(4),1 (1983).

    Google Scholar 

  38. F. Chaitin-Chatelin and V. Fraysseé,Lectures on Finite Precision Computations (Society for Industrial and Applied Mathematics, Philadelphia. 1955).

    Google Scholar 

  39. H. Goldstein,Classical Mechanics, 2nd edn. (Addison-Wesley, Reading, Massachusetts, 1980).

    Google Scholar 

  40. G. H. Hardy and E. M. Wright,An Introduction to the Theory of Numbers, 3rd edn. (Cambridge University Press, London, 1954).

    Google Scholar 

  41. E. M. Gold,Inform. Control. 10, 447 (1967).

    Google Scholar 

  42. St. Wagon,The Banach-Tarski Paradox, 2nd printing (Cambridge University Press, Cambridge, 1986).

    Google Scholar 

  43. B. W. Augenstein,Int. J. Theor. Phys. 23, 1197 (1984); “Conveiving nature — discovering reality,”J. Sci. Explor. 8, 279–282 (1994).

    Google Scholar 

  44. I. Pitowsky,Phys. Rev. Lett. 48, 1299 (1982);Phys. Rev. D 27, 2316 (1983); N. D. Mermin,Phys. Rev. Lett. 49, 1214 (1982); A. L. Macdonald,ibid. 1215 (1982); I. Pitowsky,ibid. 1216 (1982);Quantum Probability—Quantum Logic (Springer, Berlin, 1989).

    Google Scholar 

  45. H. G. Schuster,Deterministic Chaos (Physik Verlag, Weinheim 1984).

    Google Scholar 

  46. J.-P. Eckmann and D. Ruelle,Rev. Mod. Phys. 57, 617 (1985).

    Google Scholar 

  47. Tarski's theorem can be stated as follows(42):. Suppose a group G acts onA ⊂X. Then there exists a finitely-additive,G-invariant measureΜ:B(x) ↿ [0, ∞) with Μ(A) = 1 if and only ifA isnot G-paradoxical.

    Google Scholar 

  48. H. D. P. Lee,Zeno of Elea (Cambridge University Press, Cambridge, 1936; reprinted by Adolf M. Hakkert, Amsterdam, 1967).

    Google Scholar 

  49. W. McLaughlin and S. L. Miller, “An epistemological use of nonstandard analysis to answer Zeno's objections against motion,”Synthese 92, 371–384 (1992).

    Google Scholar 

  50. H. Weyl,Philosophy of Mathematics and Natural Science (Princeton University Press, Princeton, 1949).

    Google Scholar 

  51. A. Grünbaum,Modern Science and Zeno's Paradoxes, 2nd edn (Allen & Unwin, London, 1968);Philosophical Problems of Space of Time, 2nd, enlarged edn. (Reidel, Dordrecht, 1973).

    Google Scholar 

  52. J. F. Thomson, “Tasks and super-tasks,”Analysis 15, 1–3 (1954).

    Google Scholar 

  53. P. Benacerraf, “Tasks, super-tasks, and the modern eleatics,”J. Philos. 59, 765–784 (1962).

    Google Scholar 

  54. I. Pitowsky, “The physical Church-Turing thesis and physical complexity theory,”Iyyun, A Jerusalem Philosophical Quarterly 39, 81–99 (1990).

    Google Scholar 

  55. M. Hogarth, “Non-Turing computers and non-Turing computability,”PSA 1994,1, 126–138 (1994).

    Google Scholar 

  56. J. Earman and J. D. Norton, “Forever is a day: Supertasks in Pitowsky and Malament-Hogarth spacetimes,”Philos. Sci. 60, 22–42 (1993).

    Google Scholar 

  57. K. Svozil,Randomness and Undecidability in Physics (World Scientific, Signapore, 1993).

    Google Scholar 

  58. K. Svozil, “On the computional power of physical systems, undecidability, the consistency of phenomena and the practical uses of paradoxes,” inFundamental Problems in Quantum Theory: A Conference Held in Honor of Professor John A. Wheeler, D. M. Greenberger and A. Zeilinger, eds.Ann. N. Y. Acad. Sci. 755, 834–842 (1995).

  59. A. Zeilinger, private communication.

  60. O. E. Rössler, “Endophysics,” inReal Brians, Artificial Minds, J. L. Casti and A. Karlquist, eds. (North-Holland, New York, 1987), p. 25;Endophysics, Die Welt des inneren Beobachters, P. Weibel, ed. (Merwe Verlag, Berlin, 1992).

    Google Scholar 

  61. K. Svozil, “On the setting of scales for space and time in arbitrary quantized media (Lawrence Berkeley Laboratory preprint LBL-16097, May 1983);Europhys. Lett. 2, 83 (1986);Nuovo Cimento B 96, 127 (1986); see also Ref. 57.

  62. A. Zeilinger and K. Svozil,Phys. Rev. Lett. 54, 2553 (1985); K. Svozil and A. Zeilinger,Int. J. Mod. Phys. A 1, 971 (1986); K. Svozil and A. Zeilinger,Phys. Scri. T 21, 122 (1988).

    Google Scholar 

  63. K. Svozil,J. Phys. A 19, L1125 (1986); K. Svozil,J. Phys. A 20, 3861 (1987).

    Google Scholar 

  64. M. Schaller and K. Svozil,Nuovo Cimento B 109, 167 (1994);Int. J. Theor. Phys., in press.

    Google Scholar 

  65. R. Descartes, Discours de la méthode pour bien conduire sa raison et chercher la vérité dans les sciences (English translation: Discourse on the Method of Correct Reasoning and the Search for Truth in the Sciences) (1637).

  66. A. Jaffe and F. Quinn, “Theoretical mathematics: Toward a cultural synthesis of mathematical and theoretical physics,”Bull. [New Ser.] Am. Math. Soc. 29, 1–13 (1993).

    Google Scholar 

  67. E. P. Wigner, “The unreasonable effectiveness of mathematics in the natural sciences,” Richard Courant Lecture delivered at New York University, May 11, 1959 and published inCommun. Pure Appl. Math. 13, 1 (1960).

  68. Aristotle,Metaphysics, around 350 B.C., translated by W. D. Ross, e-print http://the-tech.mit.edu/Classics/Aristotle/metaphysics.txt.

  69. K. Svozil, “How real are virtual realities, how virtual is reality? The constructive re-interpretation of physical undecidability,” inComplexity, in press.

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  1. Institut für Theoretische Physik, University of Technology Vienna, Wiedner Hauptstra\e 8-10/136, A-1040, Vienna, Austria

    K. Svozil

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Svozil, K. Set theory and physics. Found Phys 25, 1541–1560 (1995). https://doi.org/10.1007/BF02055507

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