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Randomness, Unpredictability and Absence of Order: The Identification by the Theory of Recursivity of the Mathematical Notion of Random Sequence

  • Jean-Paul Delahaye
Chapter
Part of the Philosophical Studies Series book series (PSSP, volume 56)

Abstract

The theory of recursivity which was initiated by Gödel, Church, Turing and Post between 1930 and 1936 leads 30 years later to an absolute definition of randomness that seems to fulfil the main objectives stated by von Mises. The definition of random sequences by Martin-Löf in 1965 and the other works on the so-called ‘algorithmic theory of information’ by Kolmogorof, Chaitin, Schnorr and Levin (among others) may be understood as the formulation of a thesis similar to the Church-Turing’s Thesis about the notion of algorithmic calculability. Here is this new thesis we call the Martin-Löf-Chaitin’s Thesis: the intuitive informal concept of random sequences (of 0 and 1) is satisfactorily defined by the notion of Martin-Löf-Chaitin random sequences (MLC-random sequences) that is, sequences which do not belong to any recursively null set. In this paper (a short version of [Delahaye 1990]), we first recall and explain shortly the notion of MLC-random sequences; and propose afterwards a comparison between the Church-Turing’s Thesis and the Martin-Löf-Chatin’s Thesis. Our conclusion is that there is a huge similarity between the two thesis, but that today the Martin-Löf-Chaitin’s Thesis is more problematic and more complex than the Church-Turing’s Thesis.

Keywords

Random Sequence Recursive Function Soviet Math Mathematical Notion Calculable Function 
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© Springer Science+Business Media Dordrecht 1993

Authors and Affiliations

  • Jean-Paul Delahaye
    • 1
  1. 1.Lille UniversityFrance

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