All the online resources that I've seen on uncomputable numbers assume that they're all irrational. But this doesn't seem to be required by the definition. Wikipedia, for example, says that "[un]computable numbers are the real numbers that can [not] be computed to within any desired precision by a finite, terminating algorithm."
But doesn't this also describe the outputs of uncomputable functions? Busy_beaver(6) cannot be computed by any finite terminating algorithm, yet it's an integer. (Well ok, maybe the 6th is still computable and they only become uncomputable later. But that's not relevant to this question.)
Perhaps the issue here is the definition of "number". In the context of uncomputable numbers, people are interpreting Busy_beaver(6) as a statement of "we don't know what number is the result of this function", rather than a specific number. After all, whatever Busy_beaver(6) is, that number could also be computed some other way. If Busy_beaver(6) is, say, 10^20000, there's certainly a finite terminating algorithm that outputs 10^20000, even if we have no way of knowing that that number is also the result of Busy_beaver(6).
But this seems to conflict with how people talk about "numbers" in other contexts. You can define two large numbers that are the results of computable functions, (like Graham's Number or TREE(3) or whatever), and people agree that those specify a single unambiguous number, even if we don't know if two of them are equal or know their exact decimal expansion.
For that matter, how do we know that Chaitin's Constant isn't the result of any finite terminating function? The argument given on the Wikipedia page only proves that we can never compute what number is Chaitin's Constant, but it doesn't prove that Chaitin's Constant is itself not the result of any other unrelated function. (For example, how can you prove that Chaitin's Constant isn't pi/4? Maybe it is, we'd just never know.)
Clearly I'm very confused about what types of statement count as clearly defining a single number. Help?
