Quote Originally Posted by JCarter426 View Post

I mean, it's not incredibly profound, as we said before, but that's the paradox of it. If the question is whether the process is true for all reals, on the surface the answer should be no, because we know there are uncomputable numbers, so we shouldn't be able to find a process that works for them. But we can also demonstrate a process that does work if we have an oracle to confirm the answer. The conclusion, of course, is that they can't actually be thinking of an uncomputable number and can only know some finite approximation of it. But that goes against the premise of the question.
I think you're misinterpreting what it means for a number to be uncomputable. It does not mean that the number is impossible to represent. It means that the number is impossible to identify the value of from its definition.

For example, consider the two thousandth Busy Beaver number. It's a finite integer by the nature of its definition, so if someone actually knew its value and had enough time, space, and materials it would obviously be possible to write it down. At the same time, it has been proven to be uncomputable. This is not a contradiction, because it just means that no one will ever know which integer is "the two thousandth Busy Beaver number".