Notre Dame Journal of Formal Logic

Continu'ous Time Goes by Russell

Uwe Lück

Abstract

Russell and Walker proposed different ways of constructing instants from events. For an explanation of "time as a continuum," Thomason favored Walker's construction. The present article shows that Russell's construction fares as well. To this end, a mathematical characterization problem is solved which corresponds to the characterization problem that Thomason solved with regard to Walker's construction. It is shown how to characterize those event structures (formally, interval orders) which, through Russell's construction of instants, become linear orders isomorphic to a given (or, deriving, to some—nontrivial ordered) real interval. As tools, separate characterizations for each of resulting (i) Dedekind completeness, (ii) separability, (iii) plurality of elements, (iv) existence of certain endpoints are provided. Denseness is characterized to replace Russell's erroneous attempt. Somewhat minimal nonconstructive principles needed are exhibited, and some alternative approaches are surveyed.

Article information

Source
Notre Dame J. Formal Logic Volume 47, Number 3 (2006), 397-434.

Dates
First available in Project Euclid: 17 November 2006

http://projecteuclid.org/euclid.ndjfl/1163775446

Digital Object Identifier
doi:10.1305/ndjfl/1163775446

Mathematical Reviews number (MathSciNet)
MR2264708

Zentralblatt MATH identifier
1113.03012

Citation

Lück, Uwe. Continu'ous Time Goes by Russell. Notre Dame J. Formal Logic 47 (2006), no. 3, 397--434. doi:10.1305/ndjfl/1163775446. http://projecteuclid.org/euclid.ndjfl/1163775446.

References

• [1] Allen, J. F., and P. J. Hayes, "A common-sense theory of time", pp. 528--31 in Proceedings of the Ninth International Joint Conference on Artificial Intelligence, edited by A. Joshi, Kaufmann, Los Altos, 1985.
• [2] Anderson, C. A., "Russell on order in time", pp. 249--63 in Rereading Russell: Essays in Bertrand Russell's Metaphysics and Epistemology, edited by C. W. Savage and C. A. Anderson, vol. 12 of Minnesota Studies in the Philosophy of Science, University of Minnesota Press, Minneapolis, 1989.
• [3] Bell, J. L., and A. B. Slomson, Models and Ultraproducts: An Introduction, North-Holland Publishing Co., Amsterdam, 1969.
• [4] van Benthem, J. F. A. K., The Logic of Time. A Model-Theoretic Investigation into the Varieties of Temporal Ontology and Temporal Discourse, 2d edition, vol. 156 of Synthese Library, Kluwer, Dordrecht, 1991.
• [5] Birkhoff, G., Lattice Theory, 3d edition, American Mathematical Society Colloquium Publications, vol. 25. American Mathematical Society, Providence, 1973.
• [6] Bridges, D., and F. Richman, Varieties of Constructive Mathematics, vol. 97 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 1987.
• [7] Burgess, J. P., "Axioms for tense logic. II". Time periods, Notre Dame Journal of Formal Logic, vol. 23 (1982), pp. 375--83.
• [8] Burgess, J. P., "Beyond tense logic", Journal of Philosophical Logic, vol. 13 (1984), pp. 235--48.
• [9] Cantor, G., "Beiträge zur Begründung der transfiniten Mengenlehre", Mathematische Annalen, vol. 46 (1897), pp. 481--512. Translation by P. E. B. Jourdain: Contributions to the Founding of the Theory of the Transfinite Numbers, reprinted by Dover Publications, New York, 1955.
• [10] Cohen, P. J., Set Theory and the Continuum Hypothesis, W. A. Benjamin, Inc., New York-Amsterdam, 1966.
• [11] Drake, F. R., Set Theory. An Introduction to Large Cardinals, North-Holland Publishing Co., Amsterdam, 1974.
• [12] Felgner, U., Models of $\rm ZF$-Set Theory, vol. 223 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1971.
• [13] Fishburn, P., and B. Monjardet, "Norbert Wiener on the theory of measurement (1914, 1915, 1921)", Journal of Mathematical Psychology, vol. 36 (1992), pp. 165--84.
• [14] Fishburn, P. C., "Intransitive indifference with unequal indifference intervals", Journal of Mathematical Psychology, vol. 7 (1970), pp. 144--49.
• [15] Fishburn, P. C., Interval Orders and Interval Graphs. A Study of Partially Ordered Sets, Wiley-Interscience Series in Discrete Mathematics. John Wiley & Sons Ltd., Chichester, 1985.
• [16] Hamblin, C. L., "Instants and intervals", pp. 324--31 in The Study of Time, edited by J. T. Fraser, F. C. Haber, and G. H. Müller, vol. 1, Springer, Berlin, 1972. Previously published in Studium Generale, vol. 24 ( in 1971).
• [17] Hawking, S. W., A Brief History of Time: From the Big Bang to Black Holes, Bantam Books, New York, 1988.
• [18] Jech, T. J., "About the axiom of choice", pp. 345--70 in Handbook of Mathematical Logic, edited by J. Barwise, H. J. Keisler, K. Kunen, Y. N. Moschovakis, and A. S. Troelstra, vol. 90 of Studies in Logic and the Foundations of Mathematics, North-Holland Publishing Co., Amsterdam, 1977.
• [19] Kamp, H., "Events, instants and temporal reference", pp. 376--417 in Semantics from Different Points of View, edited by R. Bäuerle, C. Schwarze, and A. von Stechow, Springer, Berlin, 1979.
• [20] Kelley, J. L., General Topology, D. Van Nostrand Company, Inc., Toronto-New York-London, 1955.
• [21] Kleinknecht, R., "Zeitordnung und Zeitpunkte", Erkenntnis, vol. 54 (2001), pp. 55--75. Festschrift in honour of Wilhelm K. Essler on the occasion of his 60th birthday.
• [22] Kunen, K., Set Theory. An Introduction to Independence Proofs, vol. 102 of Studies in Logic and the Foundations of Mathematics, North-Holland Publishing Co., Amsterdam, 1980.
• [23] Lavine, S., Understanding the Infinite, Harvard University Press, Cambridge, 1994.
• [24] Lévy, A., Basic Set Theory, Springer-Verlag, Berlin, 1979.
• [25] Luce, R. D., "Semiorders and a theory of utility discrimination", Econometrica, vol. 24 (1956), pp. 178--91.
• [26] Masani, P. R., Norbert Wiener: 1894--1964, vol. 5 of Vita Mathematica, Birkhäuser Verlag, Basel, 1990.
• [27] Menger, K., Kurventheorie, Teubner, Leipzig, 1932.
• [28] Moore, G. H., Zermelo's Axiom of Choice. Its Origins, Development, and Influence, vol. 8 of Studies in the History of Mathematics and Physical Sciences, Springer-Verlag, New York, 1982.
• [29] Narens, L., "Measurement without Archimedean axioms", Philosophy of Science, vol. 41 (1974), pp. 374--93.
• [30] Narens, L., Abstract Measurement Theory, The MIT Press, Cambridge, 1985.
• [31] O'Neill, B., Semi-Riemannian Geometry. With Applications to Relativity, vol. 103 of Pure and Applied Mathematics, Academic Press Inc., New York, 1983.
• [32] Parchomenko, A. S., Was ist eine Kurve?, VEB Deutscher Verlag der Wissenschaften, Berlin, 1957.
• [33] Reinhardt, F., and H. Soeder, dtv-Atlas zur Mathematik. Tafeln und Texte, Band 1: Grundlagen, Algebra und Geometrie, Deutscher Taschenbuch Verlag, Munich, 1974.
• [34] Rosenstein, J. G., Linear Orderings, vol. 98 of Pure and Applied Mathematics, Academic Press Inc., New York, 1982.
• [35] Rubin, H., and J. E. Rubin, Equivalents of the Axiom of Choice, Studies in Logic and the Foundations of Mathematics. North-Holland Publishing Co., Amsterdam, 1970.
• [36] Russell, B., Our Knowledge of the External World, Open Court, Chicago, 1914. Later editions: 1926, Allen and Unwin; 1929, Norton.
• [37] Russell, B., "On order in time", pp. 347--63 in Logic and Knowledge. Essays 1901--1950, edited by R. C. Marsh, Allen and Unwin, London, 1956.
• [38] Schuster, P. M., "Unique existence, approximate solutions, and countable choice", Theoretical Computer Science, vol. 305 (2003), pp. 433--55.
• [39] Thomas, R., "As Time Goes By", pp. 41--42 in Jazz-Standards. Das Lexikon, edited by H.-J. Schaal, Bärenreiter, Kassel, 2002.
• [40] Thomason, S. K., "On constructing instants from events", Journal of Philosophical Logic, vol. 13 (1984), pp. 85--96.
• [41] Thomason, S. K., "Free construction of time from events", Journal of Philosophical Logic, vol. 18 (1989), pp. 43--67.
• [42] Walker, A. G., "Durées et instants", Revue Scientifique, vol. 85 (1947), pp. 131--34.
• [43] Walker, A. G., "Foundations of relativity, Parts I and II", pp. 319--35 in Proceedings of the Royal Society of Edinburgh (Section A), vol. 62, 1948.
• [44] Whitehead, A. N., and B. Russell, Principia Mathematica, vol. 3, Cambridge University Press, Cambridge, 1913.
• [45] Whitrow, G. J., The Natural Philosophy of Time, 2d edition, The Clarendon Press, New York, 1980.
• [46] Wiener, N., "A contribution to the theory of relative position", Proceedings of the Cambridge Philosophical Society, vol. 17 (1914), pp. 441--49.