That's right, we are talking about non computable numbers, so they can't actually be "specific".
Say you have a no computable number A (Chaitin's constant), and I have an uncomputable number B
Chaitin's constant but the the trillionth digit is equall to the most common digit in the infinite decimal expansion of Chaitin's constant). How can you tell if they are the same or different? In general, you can't. So what does "specific" even mean?
I'm not sure if that's relevant. You're pointing out that ordering of the reals is not computable, which is true, but ordering of the computable numbers isn't even computable. This isn't relevant to whether a symbol in a computer algebra system can refer to a particular number (whether computable or not).
Say you have a no computable number A (Chaitin's constant), and I have an uncomputable number B Chaitin's constant but the the trillionth digit is equall to the most common digit in the infinite decimal expansion of Chaitin's constant). How can you tell if they are the same or different? In general, you can't. So what does "specific" even mean?