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A fruitful way of obtaining meaningful, possibly concrete, algorithmically random numbers is to consider a potential behaviour of a Turing machine and its probability with respect to a measure (or semi-measure) on the input space of binary codes. For example, Chaitin's Ω is a well known Martin-Löf random number that is obtained by considering the halting… (More)

Although algorithmic randomness with respect to various non-uniform computable measures is well-studied, little attention has been paid to algorithmic randomness with respect to computable trivial measures, where a measure μ on 2 ω is trivial if the support of μ consists of a countable collection of sequences. In this article, it is shown that there is much… (More)

In this article, I discuss the extent to which Kolmogorov drew upon von Mises' work in addressing the problem as to why probability is applicable to events in the real world, which I refer to as the applicability problem. In particular, I highlight the role of randomness in Kolmogorov's account, and I argue that this role differs significantly from the role… (More)

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