I've been reading about Chatin's Constant, and some of the information there seems to contradict what I've heard before.
It says that the digits of Chatin's Constant can not be computed. This means that if we had a set of all the digits of Chatin's constant, let's call it $C$, there is no function that exists as a bijection between the natural numbers and $C$, because if there was, we know the natural numbers and could just plug them in to that function to compute the digits of Chatin's constant. But, this seems to go against what I've heard. On this site and others, I've been told that every set can be described by a function or rule, and therefore no sequence of numbers is truly random. If that is true, how can $C$ not be described by a function?
On the wikipedia page it says "informally represents the probability that a randomly constructed program will halt." So my theory on the answer to this question is that the constructed program already assumes true randomness, so the fact that the probability of that program halting is random should follow. But, I don't think this is true because even if the constructed program is truly random, Chatin's Constant is a real constant with digits that are also constants, and then it goes back to my above points about how if that is true it can't be random, so it seems to be a circular argument.
So, how can the digits of Chatin's constant be not describable by a function when any set of numbers can be?