No, the definition of the busy beaver problem excludes any program that never halts, regardless of how complex its behavior is.
> How would the answer for a specific integer be computable if it's impossible to determine what the value for the function of is for that integer?
Can you say precisely what you mean by “computable”? I suspect you’re using an intuitive definition that’s different from the author’s formal definition.
Then we will never be able to find the Nth beaver number for the corresponding N.
That doesn’t mean it is undefined or uncomputable. It actually has nothing to do with it. There is a computer program that prints out the number of hairs on my head. Doesn’t matter that you will never know how to write that program.
Again, uncomputable is being used in a technical sense here which is why I asked you what definition of “computable” you’re using.
> Then we will never be able to find the Nth beaver number for the corresponding N.
This is pretty clearly what people are asking about when asking if BB(6) is "uncomputable" then.
I understand (now) the point about the specific meaning of the term of art "uncomputable". If you want to speak precisely on the topic, it's not the right question to ask.
But it still seems like the question, "will we ever be able to find out the Nth beaver number", is the more interesting question.
So, we can define ZFC axioms that classify the beavers we can know together with the beavers we cannot know. So what? Then that is just skipping the interesting question. Maybe that just means for this specific problem that is not the best construction to decide to use to classify them?
I would be more interested in a classification that would assign one label to beavers we could possibly hope to calculate, and another for beavers where the calculation is impossible.
“Probably doesn’t halt” is ill defined, it either halts or it doesn’t. In either case, computability of that number is completely separate from whether we can prove that number is BB(N) or decide whether a given beaver halts. Additionally, the way the busy beaver function is defined ensures it has a defined integer output. Computability is just our ability to construct a machine which would print out that number. We can clearly do that for any integer
> How would the answer for a specific integer be computable if it's impossible to determine what the value for the function of is for that integer?
Can you say precisely what you mean by “computable”? I suspect you’re using an intuitive definition that’s different from the author’s formal definition.