> You can't say "it just takes a lot of time" if that time is > age of the universe.
1. The difference matters a WHOLE LOT when you're studying computability. If the question you're trying to answer is "Can this be computed exactly?", then the distinction is paramount.
2. Your argument here is functionally equivalent to stating that there exists a largest number since it's trivial to construct a number which would require more digits (in any base you'd like) than subatomic particles in the entire universe to represent in its fully-expanded form… so let's pick this one as the largest and stop worrying about all of that infinity hub-bub!
1. The difference matters a WHOLE LOT when you're studying computability. If the question you're trying to answer is "Can this be computed exactly?", then the distinction is paramount.
2. Your argument here is functionally equivalent to stating that there exists a largest number since it's trivial to construct a number which would require more digits (in any base you'd like) than subatomic particles in the entire universe to represent in its fully-expanded form… so let's pick this one as the largest and stop worrying about all of that infinity hub-bub!