If hypercomputation is possible, then there might be a way to express some of those uncomputable numbers. They just won't be possible with an ordinary Turing machine.
(If description is all you need, then it's already possible to describe some uncomputable numbers like Chaitin's constant. But you can't reliably list its digits on an ordinary computer.)
As for the other interpretation, "have we conclusively proven we can't reach them with an ordinary computer", IIRC, the proof that there are infinite uncomputable numbers is as follows: Consider a finitely large program that, when run, outputs the number in question. This program can be encoded as an integer - just read its (binary or source) bytes as a very large base-256 number. Since the set of possible programs is no larger than the set of integers, it's (at most) countably infinite. However, the real numbers are uncountably infinite. Thus a real number is almost never computable.
If hypercomputation is possible, then there might be a way to express some of those uncomputable numbers. They just won't be possible with an ordinary Turing machine.
(If description is all you need, then it's already possible to describe some uncomputable numbers like Chaitin's constant. But you can't reliably list its digits on an ordinary computer.)
As for the other interpretation, "have we conclusively proven we can't reach them with an ordinary computer", IIRC, the proof that there are infinite uncomputable numbers is as follows: Consider a finitely large program that, when run, outputs the number in question. This program can be encoded as an integer - just read its (binary or source) bytes as a very large base-256 number. Since the set of possible programs is no larger than the set of integers, it's (at most) countably infinite. However, the real numbers are uncountably infinite. Thus a real number is almost never computable.