Uncomputability
A real number is called computable if there is an algorithm which, given n, returns the first n digits of the number. This is equivalent to the existence of a program that enumerates the digits of the real number.
No halting probability is computable. The proof of this fact relies on an algorithm which, given the first n digits of Ω, solves Turing's halting problem for programs of length up to n. Since the halting problem is undecidable, Ω can not be computed.
The algorithm proceeds as follows. Given the first n digits of Ω and a k≤n, the algorithm enumerates the domain of F until enough elements of the domain have been found so that the probability they represent is within 2-(k+1) of Ω. After this point, no additional program of length k can be in the domain, because each of these would add 2-k to the measure, which is impossible. Thus the set of strings of length k in the domain is exactly the set of such strings already enumerated.
Read more about this topic: Chaitin's Constant