Maybe there's no such thing as a random sequence
An infinite binary sequence is deemed to be random if it has all definable properties that hold almost surely for the usual probability measure on the set of infinite binary sequences. There are only countably many such properties, so it would seem that the set of random sequences should have full measure. But in fact there might be no random sequences, because for all we know, there might be no undefinable sets.
http://arxiv.org/abs/1103.3494
It would take an infinite amount of time to truly verify any random sequence. (Probably)
