Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-05T16:29:24.721Z Has data issue: false hasContentIssue false
  • English
  • Français

The axiomatization of randomness

Published online by Cambridge University Press:  12 March 2014

Michiel van Lambalgen
Michiel van Lambalgen*
Affiliation:
Faculteit Wiskunde en Informatica, Universiteit van Amsterdam, 1018 TV Amsterdam, The Netherlands
Get access Rights & Permissions [Opens in a new window]

Abstract

We present a faithful axiomatization of von Mises' notion of a random sequence, using an abstract independence relation. A byproduct is a quantifier elimination theorem for Friedman's “almost all” quantifier in terms of this independence relation.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 55 , Issue 3 , September 1990 , pp. 1143 - 1167
Copyright
Copyright © Association for Symbolic Logic 1990

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

REFERENCES

Baldwin, J. T. [1988], Stability theory, Springer-Verlag, Berlin.Google Scholar
Bell, J. L. [1977], Boolean valued models and independence proofs in set theory, Oxford University Press, Oxford.Google Scholar
Chaitin, G. J. [1987], Algorithmic information theory, Cambridge University Press, Cambridge.CrossRefGoogle Scholar
Church, A. [1940], On the concept of a random sequence, Bulletin of the American Mathematical Society, vol. 46, pp. 46–130.CrossRefGoogle Scholar
Fine, K. [1985], Natural deduction and arbitrary objects, Journal of Philosophical Logic, vol. 14, pp. 14–57.CrossRefGoogle Scholar
Freiling, C. [1986], Axioms of symmetry: throwing darts at the real number line, this Journal, vol. 51, pp. 190–200.Google Scholar
Gaifman, H. and Snir, M. [1982], Probabilities over rich languages, randomness and testing, this Journal, vol. 47, pp. 495–548.Google Scholar
Kolmogorov, A. N. and Uspensky, V. A. [1988], Algorithms and randomness, Theory of Probability and Its Applications, vol. 32, pp. 32–389.CrossRefGoogle Scholar
Krivine, J. L. and McAloon, K. [1973], Forcing and generalized quantifiers, Annals of Mathematical Logic, vol. 5, pp. 5–199.CrossRefGoogle Scholar
van Lambalgen, M. [1987a], Von Mises definition of random sequences reconsidered, this Journal, vol. 52, pp. 725–755.Google Scholar
van Lambalgen, M. [1987b], Random sequences, Ph.D. thesis, Department of Mathematics, University of Amsterdam, Amsterdam.Google Scholar
van Lambalgen, M. and Moerdijk, I. [199?], Models for randomness (in preparation).Google Scholar
Martin-Löf, P. [1966], The definition of random sequences, Information and Control, vol. 9, pp. 9–602.CrossRefGoogle Scholar
Martin-Löf, P. [1971], Complexity oscillations in infinite binary sequences, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, vol. 19, pp. 19–225.CrossRefGoogle Scholar
Metakides, G. and Nerode, A. [1980], Recursion theory on fields and abstract dependence, Journal of Algebra, vol. 65, pp. 65–36.CrossRefGoogle Scholar
von Mises, R. [1919], Grundlagen der Wahrscheinlichkeitsrechnung, Mathematische Zeitschrift, vol. 5, pp. 5–52.Google Scholar
Oxtoby, J. C. [1980], Measure and category, Springer-Verlag, Berlin.CrossRefGoogle Scholar
Parthasaraty, T. [1972], Selection theorems and their applications, Lecture Notes in Mathematics, vol. 263, Springer-Verlag, Berlin.CrossRefGoogle Scholar
Solovay, R. M. [1970], A model of set theory in which every set of reals is Lebesgue measurable, Annals of Mathematics, ser. 2, vol. 90, pp. 1–55.CrossRefGoogle Scholar
Steinhorn, C. I. [1985a] Borel structures and measure and category logics, Model-theoretic logics (Barwise, K. J. and Feferman, S., editors), Springer-Verlag, Berlin, pp. 579–596.Google Scholar
Steinhorn, C. I. [1985b], Borel structures for first order and extended logics, Harvey Friedman's research on the foundations of mathematics (Harrington, L. A.et al., editors), North-Holland, Amsterdam, pp. 161–178.CrossRefGoogle Scholar
Troelstra, A. S. [1977], Choice sequences, Oxford University Press, Oxford.Google Scholar
Troelstra, A. S. and van Dalen, D. [1988], Constructivism in mathematics. Vols. I, II, North-Holland, Amsterdam.Google Scholar
van der Waerden, B. L., Modern algebra. Vols, I, II, Ungar, New York.Google Scholar
Welsh, D. J. A., Matroid theory, Academic Press, New York.Google Scholar