Links between physics and set theory

Chaos, Solitons and Fractals

Volumn 7, Issue 11, 1996, Pages 1761-1798

  Augenstein, Bruno W ^a  

^a RAND CORPORATION   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0042091914     PISSN: 09600779     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0960-0779(96)00055-0     Document Type: Article

References (125)

1. F. Copleston, A History of Philosophy - Volume 1: Greece and Rome. Image Books, Doubleday (1993).
2. G. 't Hooft, Questioning the answers, or stumbling upon good and bad Theories of Everything. In Physics and Our View of the World, edited by J. Hilgevoord. Cambridge (1994).
3. S. Wagon, The Banach-Tarski Paradox. Cambridge (1985).
4. M. Potter, Sets, An Introduction. Oxford (1990).
5. P. Benioff, Models of Zermelo Frankel set theory as carriers for the mathematics of physics I, II. J. Math. Phys. 17(5), 618-640 (1976).
6. S. Kleene, Mathematical Logic. Wiley (1967).
7. T. Jech, Set Theory. Academic Press (1978).
8. P. Cohen, Set Theory and the Continuum Hypothesis. Benjamin (1966).
9. G. Moore, Zermelo's Axiom of Choice - Its Origins, Development, and Influence. Springer (1982). This is one of the most readable and comprehensive books on a broad mathematical issue.
10. H. Rubin and J. Rubin, Equivalents of the Axiom of Choice. North-Holland (1963).
11. E. Zermelo, Untersuchungen über die Grundlagen der Mengenlehre. Math. Ann. 65, 261-281 (1908).
12. A. Fraenkel, Zur den Grundlagen der Cantor-Zermeloschen Mengenlehre. Math. Ann. 86, 230-237 (1922).
13. J. Dauben, Georg Cantor - His Mathematics and Philosophy of the Infinite. Harvard (1979).
14. P. Maddy, Believing the Axioms I, II. J. Symbolic Logic. 53, 481-511 and 736-764 (1988).
15. Y. Moschovakis, Notes on Set Theory. Springer (1994).
16. K. Hrbacek and T. Jech, Introduction to Set Theory. Marcel Dekker (1984).
17. A. Turing, On computable numbers, with an application to the Entscheidungsproblem. Proc. London Math. Soc. ser. 2, 42, 230-265 (1936-37); corrections, ibid. 43, 544-546 (1937).
18. E. Post, Finite combinatory processes. Formulation I. J. Symbolic Logic. 1, 103-105 (1936).
19. R. Herkin (ed.), The Universal Turing Machine: A Half-Century Survey. Oxford (1988).
20. R. Penrose, The Emperor's New Mind. Penguin Books (1991).
21. M. Davis, Computability and Unsolvability. McGraw-Hill (1958).
22. M. Pour-El and J. Richards, Computability in Analysis and Physics. Springer (1989).
23. J. König, Zum Kontinuumproblem. Math. Ann. 60, 177-180 (1905).
24. J. Barwise (ed.), Mathematical Logic. North-Holland (1977).
25. H. Feigner, Models of Set Theory. Springer (1971).
26. C. Chang and H. Keisler, Model Theory. 3rd edn. North-Holland (1990).
27. The best reference is: K. Gödel, Collected Works, Vol. II, edited by S. Feferman et al. Oxford (1990).
28. A. Kanamori, The Higher Infinite. Springer (1994). The first volume of a comprehensive treatment of large cardinals in set theory.
29. W. Sierpinski, L'hypothèse généralisée du continu et l'axiome du choix. Fund. Math. 34, 1-5 (1947).
30. S. Saunders and H. Brown, The Philosophy of Vacuum. Oxford (1991).
31. H. Casimir, On the attraction between two perfectly conducting plates. Konink. Nederl. Akad. Wetensch. B51, 793-796 (1948).
32. R. Forward, Mass Modification Experiment Definition Study, Phillips Lab. Report PL-TR-96-3004 (1996). Full comprehensive reference on experiments re vacuum fluctuations, etc.
33. R. Forward, Extracting electrical energy from the vacuum by cohesion of charged foliated conductors. Phys. Rev. B 30(4), 1700-1702 (1984).
34. H. Froning, Vacuum energy for power and propulsive flight? AIAA J. 94-3348 (1994).
35. W. Unruh, Notes of black hole evaporation. Phys. Rev. D 14, 870-892 (1976).
36. P. Davies, Scalar particle production in Schwarzschild and Rindler metrics. J. Phys. A, 8, 609-616 (1975).
37. J. Bisognano and E. Wichmann, On the duality condition for a Hermitean scalar field, J. Math. Phys. 16, 985 (1975).
38. S. Fulling and P. Davies, Radiation from a moving mirror in two dimensional space-time-conformal anomaly. Proc. Roy. Soc. A 348, 393-414 (1975).
39. B. Haisch et al., Inertia as a zero-point field Lorentz force. Phys. Rev. A 49, 678-694 (1994).
40. H. Puthoff, Gravity as a zero-point fluctuation force. Phys. Rev. A 39(5), 2333-2342 (1988).
41. I. Lawrie, A Unified Grand Tour of Theoretical Physics. Inst. of Phys. Pub. (1994).
42. A. Vilenkin, Phys. Rev. D 27, 2848 (1983).
43. A. Linde, Inflation and Quantum Cosmology. Academic Press (1990).
44. J. Hartle and S. Hawking, Phys. Rev. D 28, 2906 (1983).
45. M. Gell-Mann and J. Hartle, Quantum mechanics in the light of quantum cosmology. In Complexity, Entropy and the Physics of Information, SFI Studies, Vol. 8, edited by W. Zurek. Addison-Wesley (1990).
46. H. Duff and C. Isham, Quantum Structure of Space and Time. Cambridge (1982).
47. A. Merzabekian and G. Vilkovisky, Gravitational waves generated by the vacuum stress. Phys. Rev. Lett. 75(22), 3974-3976 (1995).
48. R. Sachs, in Relativity, Groups and Topology, edited by C. DeWitt and B. S. DeWitt. Gordon and Breach (1964).
49. A. Wightman, Hilbert's sixth problem. In Mathematical Developments Arising from the Hilbert Problems, edited by F. Browder. Proc. Sym. in pure Mathematics, Vol. XXVIII, Parts 1 and 2, ASM (1976).
50. D. Hilbert, Begründung der kinetischen Gastheorie. Math. Ann. 71, 562-577 (1912).
51. D. Hilbert, Begründung der Elementaren Strahlungstheorie. In Gesam. Abhl., Vol. III, Springer (1970).
52. D. Hilbert, Die Grundlagen der Physik. In Gesam. Abhl. Vol. III, Springer (1970).
53. D. Hilbert, J. von Neumann and L. Nordheim, Über die Grundlagen der Quantenmechanik. Math. Ann. 98, 1-30 (1927).
54. J. von Neumann, Mathematical Foundations of Quantum Mechanics, English translation. Princeton (1955). Originally published in 1927.
55. D. Hilbert, Über das Unendliche. Math. Ann. 95, 161-190 (1926); Die Grundlagen der Mathematik. Abh. Math. Sem. Univ. Hamburg 6, 65-85 (1928).
56. D. Hilbert, Über das Unendliche. Math. Ann. 95, 161-190 (1926); Die Grundlagen der Mathematik. Abh. Math. Sem. Univ. Hamburg 6, 65-85 (1928).
57. J. von Neumann, Eine Axiomatisierung der Mengenlehre. J. Reine. Angew. Math. 154, 219-240 (1925); Die Axiomatisierung der Mengenlehre. Math. Zeit. 27, 669-752 (1928).
58. J. von Neumann, Eine Axiomatisierung der Mengenlehre. J. Reine. Angew. Math. 154, 219-240 (1925); Die Axiomatisierung der Mengenlehre. Math. Zeit. 27, 669-752 (1928).
59. E. Scheibe, On the mathematical overdetermination of physics. In Philosophy, Mathematics and Modern Physics, edited by E. Rudolph and I. Stamatescu. Springer (1994). A useful overview volume.
60. G. Ludwig, An Axiomatic Basis for Quantum Mechanics. Springer (1990).
61. J. Synge, Classical dynamics. In Vol. III/1, Encyclopedia of Physics, edited by S. Flügge. Springer (1960).
62. C. Truesdell and R. Toupin, The Classical field theories. In Vol. III/1, Encyclopedia of Physics, edited by S. Flügge. Springer (1960).
63. See e.g. L. Landau and E. Liftshitz, Quantum Mechanics - Non-Relativistic Theory. Pergamon Press (1958).
64. A. Fraenkel, Der Begriff "definit" und die Unabhängigkeit des Answahlsaxioms. Sitz. Pr. Ak. Wiss. Phys.-Math. Klasse 253-257 (1922). This analogy is also noted in G. Temple, 100 Years of Mathematics. Springer (1981).
65. A. Fraenkel, Der Begriff "definit" und die Unabhängigkeit des Answahlsaxioms. Sitz. Pr. Ak. Wiss. Phys.-Math. Klasse 253-257 (1922). This analogy is also noted in G. Temple, 100 Years of Mathematics. Springer (1981).
66. A. Parker-Rhodes, The Theory of Indistinguishables. Synthese Library, Vol. 15, D. Reidel (1981).
67. G. Börner, The Early Universe - Facts and Fiction, 3rd edn. Springer (1993).
68. S. Weinberg, Rev. Mod. Phys. 61, 1 (1989).
69. H. Wang. The concept of set. In Philosophy of Mathematics, 2nd edn, edited by P. Benacerraf and H. Putnam. Cambridge (1985).
70. L. Brown (ed.), Renormalization from Lorentz to Landau (and Beyond). Springer (1993).
71. A. Tarski, Sur quelques theorems qui equivalent à l'axiome du choix. Fund. Math. 5, 147-154 (1924).
72. See e.g. K. Stroyan, Infinitesimal analysis of curves and surfaces. In ref. [24].
73. J. Conway, On Numbers and Games. Academic Press (1976).
74. E. Hewitt and K. Stromberg, Real and Abstract Analysis. Springer (1965).
75. K. Devlin, Fundamentals of Contemporary Set Theory. Springer (1979).
76. T. Jech and A. Sochor, Applications of the θ-Model. Acad. Pol. Sci. 14, 351-355 (1966).
77. H. Lauchli, Auswahlaxiom in der Algebra. Comment. Math. Helv. 37, 1-18 (1962).
78. S. Weinberg, High-energy behavior in quantum field theory. Phys. Rev. 118(3), 838-849 (1960).
79. See Ref. [71].
80. R. Rucker, One of Georg Cantor's speculations on physical infinities. Spec. Sci. Tech. 1, 419-421 (1978).
81. R. Rucker, Infinity and the Mind. Birkhauser (1982).
82. See e.g. B. DeWitt and N. Graham, The Many-Worlds Interpretation of Quantum Mechanics. Princeton (1973). Splitting/rebirth of the world is loosely prefigured in Indian philosophy.
83. W. Beyer, J. Mycielski and G.-C. Rota (eds), Stanislaw Ulam: Sets, Numbers, and Universes. MIT Press (1974).
84. N. Cooper (ed.), From Cardinals to Chaos. Cambridge (1989).
85. M. Kac and S. Ulam, Mathematics and Logic. Dover (reprint 1992).
86. R. Solovay, A model of set theory in which every set of reals is Lebesgue measurable. Ann. Math. 92, 1-56 (1970).
87. J. Shepherdson, Inner models for set theory. J. Symbolic Logic 16, 161-190 (1951); 17, 225-237 (1952); 18, 145-167 (1953).
88. J. Shepherdson, Inner models for set theory. J. Symbolic Logic 16, 161-190 (1951); 17, 225-237 (1952); 18, 145-167 (1953).
89. J. Shepherdson, Inner models for set theory. J. Symbolic Logic 16, 161-190 (1951); 17, 225-237 (1952); 18, 145-167 (1953).
90. See e.g. G. Chaitin, A New Version of Algorithmic Information Theory, Santa Fe Institute Lectures, April 1995, and references therein.
91. A. Kolmogorov, Logical basis for information theory and probability theory. IEEE Trans. Inf. Theory 14, 662-664 (1968).
92. M. Rasetti (ed.), Special Issue of Chaos, Solitons and Fractals 5(2), 000-000 (1995). See especially articles by M. Rasetti, D. Ornstein and G. Chaitin.
93. W. Boos, Metamathematical quantum theory I. J. Symbolic Logic, Abstracts of Papers, New York City, December 1987.
94. See e.g. the following and references therein: M. El Naschie, On conjugate complex time and information in relativistic quantum theory. Chaos, Solitons & Fractals 5(8), 1551-1555 (1995); Dimensions and Cantor spectra. Chaos, Solitons & Fractals 4(11), 2121-2132 (1994); Is quantum space a random Cantor set with a golden mean dimension at the core? Chaos, Solitons & Fractals 4(2), 177-179 (1994).
95. See e.g. the following and references therein: M. El Naschie, On conjugate complex time and information in relativistic quantum theory. Chaos, Solitons & Fractals 5(8), 1551-1555 (1995); Dimensions and Cantor spectra. Chaos, Solitons & Fractals 4(11), 2121-2132 (1994); Is quantum space a random Cantor set with a golden mean dimension at the core? Chaos, Solitons & Fractals 4(2), 177-179 (1994).
96. See e.g. the following and references therein: M. El Naschie, On conjugate complex time and information in relativistic quantum theory. Chaos, Solitons & Fractals 5(8), 1551-1555 (1995); Dimensions and Cantor spectra. Chaos, Solitons & Fractals 4(11), 2121-2132 (1994); Is quantum space a random Cantor set with a golden mean dimension at the core? Chaos, Solitons & Fractals 4(2), 177-179 (1994).
97. R. Mauldin and S. Williams, Random recursive construction. Trans. Am. Math. Soc. 295, 325-346 (1986).
98. F. Hausdorff, Grundzüge der Mengenlehre. Veit, Leipzig (1914).
99. S. Banach and A. Tarski, Sur la decomposition des ensembles de points en parties respectivement congruents. Fund. Math. 6, 244-277 (1924).
100. J. von Neumann, Zur allgemeinen theorie des masses. Fund. Math. 13, 73-116 (1928).
101. R. Robinson, On the decomposition of spheres. Fund. Math. 34, 246-260 (1947).
102. R. Dougherty and M. Foreman. Banach-Tarski decompositions using sets with the property of Baire. J. Am. Math. Soc. 7(1), 75-124 (1994).
103. H. Croft, R. Guy and K. Falconer, Unsolved Problems in Geometry. Springer (1991).
104. W. Sierpinski, Sur le paradoxe se MM. Banach et Tarski. Fund. Math. 33, 229-234 (1945).
105. J. Mycielski, On the paradox of the sphere. Fund. Math. 43, 348-355 (1955).
106. W. Sierpinski, Congruence of Sets. Chelsea (1953). This is probably the most easily readable introduction to the notions essential for BTT.
107. M. El Naschie, Banach-Tarski Theorem and Cantorian micro space-time. Chaos, Solitons & Fractals 5(8), 1503-1508 (1995).
108. D. Finkelstein, Finite Physics. In ref. [19].
109. L. Nottale, Fractal Space-time and Micro Physics. World Scientific (1993).
110. G. Ord, Classical analogy of quantum phased Int. J. Theor. Phys. 31, 1177-1195 (1992).
111. I. Pitowsky, Deterministic model of spin statistics. Phys. Rev. D 27(10), 2316-2326 (1983).
112. D. Malament, Critical notice - Pitowsky's quantum probability quantum logic. Philos. Sec. 59(2), 300-320 (1992).
113. B. Augenstein, Hadron physics and transfinite set theory. Int. J. Theor. Phys. 23(12), 1197-1205 (1984); Speculative model of some elementary particle phenomena. Spec. Sci. Tech. 17(1), 21-26 (1994).
114. B. Augenstein, Hadron physics and transfinite set theory. Int. J. Theor. Phys. 23(12), 1197-1205 (1984); Speculative model of some elementary particle phenomena. Spec. Sci. Tech. 17(1), 21-26 (1994).
115. D. Perkins, Introduction to High Energy Physics. Addison-Wesley (1982).
116. P. Maddy, Realism in Mathematics. Oxford (1992).
117. C. Chihara, Constructibility and Mathematical Existence. Oxford (1991).
118. J. Gribbin, Schrödinger's Kittens and the Search for Reality - Solving the Quantum Mysteries. Little (1995).
119. P. Davies, The Mind of God. In ref. [2].
120. J. Barrow, Theories of Everything. In ref. [2].
121. J. Casti, Searching for Certainty. Morrow, New York (1990).
122. It would be fascinating to undertake an annotated full compilation of allusions in the world literature to notions of infinity and reality, as is hinted at by even fragmentary collections such as those of J. Borges in Other Inquisitions 1937-1952. University of Texas (1984).
123. F. Wilczek, A crack in the standard model? Nature 380, 19-20 (1996). These transverse energies are of course those of the distinctive collimated streams of particles ('jets') which emerge at large energies and angles from PP̄ collisions, and should not be confused with the energies at which the basic P and P particle streams collide.
124. k of n. The analogy with Stage 1 of AC leaps to the eye (circumstances seem devisable for Stage 2 also). Identifying physical mechanisms driving this mysterious reduction is controversial [20]; but a further association of QM and AC lies in von Neumann's brave initial introduction [54] of human consciousness as a central part of QM and state reduction, and the counterpart discussions of consciousness and human brain psychology in early considerations of AC [9]. (Meanwhile, the Fermilab work is available: F. Abe et al., Inclusive jet cross section in P̄P collisions at √s = 1.8 TeV, Phys. Rev. Lett. 77(3), 438-443 (1996).
125. k of n. The analogy with Stage 1 of AC leaps to the eye (circumstances seem devisable for Stage 2 also). Identifying physical mechanisms driving this mysterious reduction is controversial [20]; but a further association of QM and AC lies in von Neumann's brave initial introduction [54] of human consciousness as a central part of QM and state reduction, and the counterpart discussions of consciousness and human brain psychology in early considerations of AC [9]. (Meanwhile, the Fermilab work is available: F. Abe et al., Inclusive jet cross section in P̄P collisions at √s = 1.8 TeV, Phys. Rev. Lett. 77(3), 438-443 (1996).