TITLE: Random and non-computable numbers
QUESTION [5 upvotes]: Let $\alpha \in (0,1): \quad \alpha=0.a_1a_2\cdots a_n \cdots \quad$ where the $a_n$ are numbers generated by a physical generator of genuinely random numbers (if it exists). Than it seems that $\alpha$ should be a noncomputable number. Is this statement true? I suppose yes, because we know that a sequence of random numbers cannot be generated by a Turing machine.
If this is true than we can ask if all non computable numbers can be generated by some extension of this procedure, e.g. using two random number generators that gives two numbers $n$ and $a_n$, and inserting $a_n$ in the position $n$ of the decimal part of a computable number.
Obviously this is not a practical process to write uncomputable numbers, but the sense of the question is if there exists a correspondence between sets of random numbers and non-computable numbers.

REPLY [3 votes]: There are only countably many Turing machines, and thus countably many sequences that they generate.  For any iid sequence $X_n$ of random variables that is not  almost surely constant, the probability of
a particular sequence of values is $0$, and thus by countable additivity the probability of a Turing-computable sequence is $0$.