  Both plain and prefix-free Kolmogorov randomness provide satisfactory accounts of the randomness of finite sequences. One difficulty arises, as I suggested earlier, when we attempt to extend plain Kolmogorov randomness to the case of infinite sequences in the most obvious way, that is, by defining an infinite sequence as Kolmogorov random iff all finite initial segments are Kolmogorov random. It would then turn out that no infinite sequence is random. Why? Because of the following theorem, which shows that there is no sequence such that all of its initial segments are random: