R Hansens says: >That is why I came up with the decimal digits approach.
You mean this?
(R Hansen) >Given a hypothetical process that selects real numbers entirely at random, the numbers thus selected are always of the "undeterminable" type. Essentially, think of a number with an infinite and entirely random sequence of decimal digits. Determinable numbers can have an infinite sequence of decimal digits, but the digits are never entirely random, since they are constrained by the function that creates the number.
That's a nice informal description of the difference between a computable number and a computable one.
But -- now describe a process that "selects" that non-computable number. The best I could come up would be a Geiger-counter driven digit generator. But it would never get to the end of that first number. That "dart" would never really reach the board! So saying "pick a random real ...." really hides quite a bit of sisyphean effort.