Hostname: page-component-76fb5796d-zzh7m Total loading time: 0 Render date: 2024-04-26T01:29:25.979Z Has data issue: false hasContentIssue false
  • English
  • Français

ORTHOMODULAR-VALUED MODELS FOR QUANTUM SET THEORY

Published online by Cambridge University Press:  05 June 2017

MASANAO OZAWA
MASANAO OZAWA*
Affiliation:
Graduate School of Informatics, Nagoya University
*
*GRADUATE SCHOOL OF INFORMATICS NAGOYA UNIVERSITY CHIKUSA-KU, NAGOYA 464-8601, JAPAN E-mail: ozawa@is.nagoya-u.ac.jp
Get access Rights & Permissions [Opens in a new window]

Abstract

In 1981, Takeuti introduced quantum set theory by constructing a model of set theory based on quantum logic represented by the lattice of closed linear subspaces of a Hilbert space in a manner analogous to Boolean-valued models of set theory, and showed that appropriate counterparts of the axioms of Zermelo–Fraenkel set theory with the axiom of choice (ZFC) hold in the model. In this paper, we aim at unifying Takeuti’s model with Boolean-valued models by constructing models based on general complete orthomodular lattices, and generalizing the transfer principle in Boolean-valued models, which asserts that every theorem in ZFC set theory holds in the models, to a general form holding in every orthomodular-valued model. One of the central problems in this program is the well-known arbitrariness in choosing a binary operation for implication. To clarify what properties are required to obtain the generalized transfer principle, we introduce a class of binary operations extending the implication on Boolean logic, called generalized implications, including even nonpolynomially definable operations. We study the properties of those operations in detail and show that all of them admit the generalized transfer principle. Moreover, we determine all the polynomially definable operations for which the generalized transfer principle holds. This result allows us to abandon the Sasaki arrow originally assumed for Takeuti’s model and leads to a much more flexible approach to quantum set theory.

Keywords

03E4003E7003E7503G1206C1546L6081P10quantum logicset theoryBoolean-valued modelsforcingtransfer principleorthomodular latticescommutatorimplicationvon Neumann algebras
Type
Research Article
Information
The Review of Symbolic Logic , Volume 10 , Issue 4 , December 2017 , pp. 782 - 807
Copyright
Copyright © Association for Symbolic Logic 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

BIBLIOGRAPHY

Araki, H. (2000). Mathematical Theory of Quantum Fields. Oxford: Oxford University Press.Google Scholar
Bell, J. L. (2005). Set Theory: Boolean-Valued Models and Independence Proofs (third edition). Oxford: Oxford University Press.CrossRefGoogle Scholar
Berberian, S. K. (1972). Baer *-Rings. Berlin: Springer.CrossRefGoogle Scholar
Birkhoff, G. & von Neumann, J. (1936). The logic of quantum mechanics. Annals of Mathematics, 37, 823–843.CrossRefGoogle Scholar
Bruns, G. & Kalmbach, G. (1973). Some remarks on free orthomodular lattices. In Schmidt, J., editor. Proceedings of the Lattice Theory Conference, Houston, TX, pp. 397–408.Google Scholar
Chevalier, G. (1989). Commutators and decompositions of orthomodular lattices. Order, 6, 181–194.CrossRefGoogle Scholar
Cohen, P. J. (1963). The independence of the continuum hypothesis I. Proceedings of the National Academy of Sciences of the United States of America, 50, 1143–1148.CrossRefGoogle Scholar
Cohen, P. J. (1966). Set Theory and the Continuum Hypothesis. New York: Benjamin.Google Scholar
Dirac, P. A. M. (1958). The Principles of Quantum Mechanics (fourth edition). Oxford: Oxford University Press.CrossRefGoogle Scholar
Fourman, M. P. & Scott, D. S. (1979). Sheaves and logic. In Fourman, M. P., Mulvey, C. J., and Scott, D. S., editors. Applications of Sheaves. Lecture Notes in Mathematics, Vol. 753. Berlin: Springer, pp. 302–401.CrossRefGoogle Scholar
Georgacarakos, G. N. (1979). Orthomodularity and relevance. Journal of Philosophical Logic, 8, 415–432.CrossRefGoogle Scholar
Grayson, R. J. (1979). Heyting-valued models for intuitionistic set theory. In Fourman, M. P., Mulvey, C. J., and Scott, D. S., editors. Applications of Sheaves. Lecture Notes in Mathematics, Vol. 753. Berlin: Springer, pp. 402–414.CrossRefGoogle Scholar
Hardegree, G. M. (1981). Material implication in orthomodular (and Boolean) lattices. Notre Dame Journal of Formal Logic, 22, 163–182.CrossRefGoogle Scholar
Herman, L., Marsden, E. L., & Piziak, R. (1975). Implication connectives in orthomodular lattices. Notre Dame Journal of Formal Logic, 16, 305–328.CrossRefGoogle Scholar
Husimi, K. (1937). Studies on the foundation of quantum mechanics I. Proceedings of the Physico-Mathematical Society of Japan, 19, 766–778.Google Scholar
Johnstone, P. T. (1977). Topos Theory. London: Academic.Google Scholar
Kalmbach, G. (1983). Orthomodular Lattices. London: Academic.Google Scholar
Kotas, J. (1967). An axiom system for the modular logic. Studia Logica, 21, 17–38.CrossRefGoogle Scholar
Marsden, E. L. (1970). The commutator and solvability in a generalized orthomodular lattice. Pacific Journal of Mathematics, 33, 357–361.CrossRefGoogle Scholar
Ozawa, M. (2005). Perfect correlations between noncommuting observables. Physics Letters A, 335, 11–19.CrossRefGoogle Scholar
Ozawa, M. (2006). Quantum perfect correlations. Annals of Physics, 321, 744–769.Google Scholar
Ozawa, M. (2007). Transfer principle in quantum set theory. Journal of Symbolic Logic, 72, 625–648.CrossRefGoogle Scholar
Ozawa, M. (2016). Quantum set theory extending the standard probabilistic interpretation of quantum theory. New Generation Computing, 34, 125–152.CrossRefGoogle Scholar
Pulmannová, S. (1985). Commutators in orthomodular lattices. Demonstratio Mathematica, 18, 187–208.CrossRefGoogle Scholar
Sasaki, U. (1954). Orthocomplemented lattices satisfying the exchange axiom. Journal of Science of the Hiroshima University: Series A, 17, 293–302.Google Scholar
Scott, D. & Solovay, R. (1967). Boolean-valued models for set theory. Unpublished manuscript for Proceedings of AMS Summer Institute on Set Theory. Los Angeles: University of California, 1967.Google Scholar
Takeuti, G. (1981). Quantum set theory. In Beltrametti, E. G. and van Fraassen, B. C., editors. Current Issues in Quantum Logic. New York: Plenum, pp. 303–322.CrossRefGoogle Scholar
Takeuti, G. & Zaring, W. M. (1973). Axiomatic Set Theory. New York: Springer.CrossRefGoogle Scholar
Titani, S. (1999). A lattice-valued set theory. Archive for Mathematical Logic, 38, 395–421.CrossRefGoogle Scholar
Titani, S. & Kozawa, H. (2003). Quantum set theory. International Journal of Theoretical Physics, 42, 2575–2602.CrossRefGoogle Scholar
Urquhart, A. (1983). Review. Journal of Symbolic Logic, 48, 206–208.CrossRefGoogle Scholar
von Neumann, J. (1955). Mathematical Foundations of Quantum Mechanics. Princeton, NJ: Princeton University Press. [Originally published: Mathematische Grundlagen der Quantenmechanik (Springer, Berlin, 1932)].Google Scholar