A model for spacetime: The role of interpretation in some Grothendieck topoi

Foundations of Physics

Volumn 36, Issue 7, 2006, Pages 1070-1098

  Król, Jerzy ^a  

^a UNIVERSITY OF SILESIA   (Poland)

Author keywords

Functional measures; Intuitionism; Quantum mechanics; Spacetime structure; Topos theory

Indexed keywords

EID: 33749329154     PISSN: 00159018     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10701-006-9052-1     Document Type: Article

References (41)

1. T. Asselmeyer, "Generation of source terms in general relativity by differential structures," Class. Quant. Grav. 14, 749-758 (1997).
2. T. Asselmeyer-Maluga and C. H. Brans, "Cosmological anomalies and exotic smoothness structures," Gen. Rel. Grav. 34(10), 1767-1771 (2002).
3. T. Asselmeyer-Maluga and H. Rosé, "Differential structures-the geometrization of quantum mechanics," e-print gr-qc/0511089.
4. M. Barr, "Toposes without points," J. Pure Appl. Alg. 5, 265 (1974).
5. J. L. Bell, Boolean-valued Models and Independence Proofs in Set Theory, 2nd edn. (Oxford University, Oxford, 1985).
6. J. L. Bell, "Orthospaces and quantum logic," Found. Phys. 15(12), 1179-1202 (1985).
7. J. L. Bell, "From absolute to local mathematics," Synthese 69, 409-426 (1986).
8. P. Benioff, "Models of ZF set theory as carriers for the mathematics of physics I, II," J. Math. Phys. 19, 618, 629 (1976).
9. P. Benioff, "Language is physical," Quantum Inf. Proc. 1, 4495, e-print quant-ph/0210211 (2002).
10. P. Benioff, "Towards a coherent theory of physics and mathematics," Found. Phys.32, 989-1029 (2002).
11. C. H. Brans and D. Randall, "Exotic differentiable structures and general relativity," Gen. Rel. Grav. 25, 205 (1993).
12. S. Carlip, "Quantum gravity: a progress report," Rept. Prog. Phys. 64, 885 (2001).
13. M. Davis, "A relativity principle in quantum mechanics," Int. J. Theor. Phys. 16(11), 867-874 (1976).
14. C. de Witt-Morette, "Feynman path integrals. I. Linear and affine transformations, II. The Feynman Green's functions," Comm. Math. Phys. 37, 63 (1974).
15. M. P. Fourman and J. M. E. Hyland, Sheaf Models for Analysis (Lecture Notes in Mathematics 753), (Springer, Berlin, 1979), pp. 302-401.
16. A. K. Guts, "Axiomatic relativity theory," Uspekhi Mat. Nauk 37(2), 39-79 (1982);
17. Russian Math. Survey 37(2), 41-89 (1982).
18. A. K. Guts and E. B. Grinkevich, "Toposes in general theory of relativity," e-print gr-qc/9610073 (1996).
19. A. Heyting, Intuitionism, An Introduction, 3rd edn. (North Holland, New York, 1971).
20. C. J. Isham, "Some possible role for topos theory in quantum theory and quantum gravity," Found. Phys. 30(10), 1707-1735 (2000).
21. C. J. Isham and J. Butterfield, "A topos perspective on the Kochen-Specker theorem: I. Quantum states as generalized valuations," Int. J. Theor. Phys. 37, 2669-2733 (1998).
22. C. J. Isham and J. Butterfield, "A topos perspective on the Kochen-Specker theorem: II. Conceptual aspects and classical analogues," Int. J. Theor. Phys. 38, 827-859 (1999).
23. T. Jech, Set Theory (Springer, Berlin, Heildelberg, New York, 2003).
24. P. T. Johnstone, Sketches of an Elephant. A Topos Theory Compendium (Clarendon, Oxford, 2002).
25. M. Kaku, Quantum Field Theory. A Modern Introduction (Oxford University Press, New York, Oxford, 1993).
26. J. Król, "Formal languages and model theoretic perspectives in physics," Acta Phys. Pol. B 32(11), 3855 (2001).
27. J. Król, "Background independence in quantum gravity and forcing constructions," Found Phys. 34(3), 361-403 (2004).
28. J. Król, "Exotic smoothness and noncommutative spaces. The model-theoretical approach," Found. Phys. 34(5), 843-869 (2004).
29. J. Król, "Model theory and the AdS/CFT correspondence," e-print hep-th/0506003 (2005).
30. J. Lambek and P. J. Scott, Introduction to Higher Order Categorical Logic (Cambridge University, Cambridge, 1994).
31. F. W. Lawvere, "Continuously Variable Sets; Algebraic Geometry=Geometric Logic" Logic Coll. 73, 135 (1975).
32. S. Mac Lane and I. Moerdijk, Sheaves in Geometry and Logic. A First Introduction to Topos Theory (Springer, New York, 1992).
33. I. Moerdijk and G. E. Reyes, Models for Smooth Infinitesimal Analysis (Springer, New York, 1991).
34. H. Pfeiffer, "Quantum general relativity and the classification of smooth manifolds," e-print gr-qc/0404088 (2004).
35. A. Scedrov, "Diagonalization of continuous matrices as a representation of intuitionistic reals," Ann. P. Appl. Logic 30, 201 (1986).
36. D. Scott, "Boolean models and nonstandard analysis," in Applications of Model Theory to Algebra, Analysis, and Probability, W. A. J. Luxemburg, ed. (Holt, Reinehart, Winston, New York, 1969).
37. R. Sikorski and H. Rasiowa, The Mathematics of Metamathematics (PWN, Warszawa, 1963).
38. 4's with few symmetries," Int. J. Mod. Phys. D10, 311 (2001).
39. G. Takeuti, "Two applications of logic to mathematics," Math. Soc. Jpn 13, Kano Memorial Lee. 3 (1978).
40. G. Takeuti, Boolean Valued Analysis, Fourman, Malvey, Scott, eds. (Lect. Notes Math. 753, Applications of Sheaves, Springer, Heildelberg, 1979), p. 714.
41. G. Takeuti, "Quantum logic and quantization," in Foundations of Quantum Mechanics in the Light of New Technologies (Tokyo, 1983), (Phys. Soc. Japan, Tokyo, 1984) pp. 256-260.