Breakdown of Predictability: An Investigation into the Nature of Singularities

  • K. Tahir Shah


I met the late Professor Wolfgang Yourgrau in 1977 during my summer visit to ICTP, Trieste. After some trivial conversation he asked me what kind of physics I do. My answer was sort of a lament: “I am afraid of physics” I told him. Whenever my confusion reaches a critical point I take refuge in pure mathematics. On the other hand, I want to understand nature and do not want to abandon physics. I do not recall his exact advice, but it was a pretty long discussion about science, philosophy, and mathematics. He persuaded me to devote my time to questions deeper than just computations and proofs. I am grateful for his gift of persuasion which pushed me towards this effort. I am also thankful to Professor Abdus Salam for his open invitation to all of us from developing countries to come to the Trieste Centre and make use of it in our intellectual endeavors. When Professor van der Merwe invited me to contribute to Yourgrau’s memorial volume I felt greatly honored and decided to contribute an enquiry into the nature of singularities from the foundational point of view. I believe it is very close to Wolfgang’s spirit; however, it is a modest enterprise of a nonexpert, as I am neither a logician nor a philosopher.


Number System Nonstandard Analysis Nonstandard Model Primitive Notion Inaccessible Cardinal 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References and Notes

General References on Logic, Sets, and Foundations

  1. 1.
    D. W. Barnes and J. M. Mack ,An Aigebraic Introduction to Mathematical Logic (Springer-Verlag, Berlin, 1975).Google Scholar
  2. 2.
    N. Bourbaki, Theory of Sets (Hermann, Paris, 1968).Google Scholar
  3. 3.
    P. Suppes, introduction to Logic (Reinhold-Van Nostrand, New York, 1957).Google Scholar
  4. 4.
    E. W. Beth, The Foundations of Mathematics (Harper, New York, 1966).Google Scholar
  5. 5.
    P. J. Cohen, Set Theory and the Continuum Hypothesis (Benjamin ,Reading, Massachusetts, 1966).Google Scholar
  6. 6.
    A. A. Fraenkel and Y. Bar-Hillel, Foundations of Set Theory (North-Holland, Amsterdam, 1958).Google Scholar
  7. 7.
    W. Yourgrau and A. D. Breck, Physics, Logic, and History (Plenum Press, New York, 1970).CrossRefGoogle Scholar
  8. 8.
    H. Margenau, The Nature of Physical Reality (McGraw-Hill, New York, 1950).Google Scholar

Physics, Singularities, and Related Ideas

  1. 9.
    N. N. Bogoliubov, Introduction to Axiomatic Quantum Field Theory. (Benjamin, Reading, Massachusetts, 1975).Google Scholar
  2. 10.
    C. Misner, K. Thorne, and J. Wheeler, Gravitation (Freeman and Co., San Francisco, 1973).Google Scholar
  3. 11.
    K. Tahir Shah and W. Yourgrau, ICTP, Trieste, Internal Report IC/78/105 (unpublished).Google Scholar
  4. 12.
    K. Tahir Shah, Found. Phys. 9, 271 (1979).CrossRefGoogle Scholar
  5. 13.
    F. Destefano and K. Tahir Shah, ICTP, Trieste, Internal Report IC/79/120.Google Scholar

Nonstandard Models and Categories

  1. 14.
    A. Robinson, Non-Standard Analysis (North-Holland, Amsterdam, 1970).Google Scholar
  2. 15.
    P. J. Kelemen and A. Robinson, J. Math. Phys. 13, 1870 (1972).CrossRefGoogle Scholar
  3. 16.
    A. Voros, J. Math. Phys. 14, 292 (1973)CrossRefGoogle Scholar
  4. 16a.
    M. O. Farrukh, ibid. 16, 177 (1975).Google Scholar
  5. 17.
    E. Nelson, “Internal Set Theory”, Bull. AMS 83, 1165 (1977).CrossRefGoogle Scholar
  6. 18.
    J. Tarski and Ph. Blanchard, Acta Phys. Austrica 49, 129 (1978).Google Scholar
  7. 19.
    W. A. Luxemburg, Applications of Model Theory to Aigebra, Analysis and Probability (Holt, Rinehart, and Winston, New York, 1959).Google Scholar
  8. 20.
    F. W. Lawvere, Proc. Symp. Pure Math. Vol. XVIII, AMS (1970).Google Scholar
  9. 21.
    A. Kuck and G. C. Wraith, Lect. Notes No. 30, Aarhus Universitet, Denmark (1971).Google Scholar
  10. 22.
    I. MacLane, Proc. Symp. Pure Math. XVIII, AMS (1970), p. 231.Google Scholar
  11. 23.
    F. W. Lawvere, “Variable Quantities and Variable Structures in Topology,” in Algebra, Topology and Category Theory ,A. Heller and M. Tierney, editors (Academic Press, New York, 1976).Google Scholar
  12. 24.
    D. A. Clarke, Hierarchies and Predicates of Finite Types ,AMS, Mem. 51 (1964).Google Scholar
  13. 25.
    A. Levy, Hierarchy of Formulas in Set Theory ,AMS, Mem. 57 (1965).Google Scholar
  14. 26.
    J. Keisler, Actes Congr. Inst. Math. 1, 141 (1970).Google Scholar

Copyright information

© Plenum Press, New York 1983

Authors and Affiliations

  • K. Tahir Shah
    • 1
  1. 1.International Centre for Theoretical PhysicsTriesteItaly

Personalised recommendations