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This is difficult to frame, but my question is if there are any finite integers that are so large that they are uncomputable in principle. Meaning that they cannot be handled by mathematics even in theory. So is there some kind of limit in abstract math where a positive number is so large that it is no longer considered encodable by symbols, even ignoring the lack of resources in the physical world to write it down. Like a fuzzy boundary where analysis isn't really sufficient to talk about them. Or are all finite numbers ultimately considered within the realm of consideration at least in an abstract sense?

Asaf Karagila
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Juel Herbranson
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    Integers are always computable according to the definition of "computable". They always have a finite decimal expansion that can , in principle , be determined. "Computable" is not meant in a practical sense. And in fact, we would run in a problem if we would try to find the largest (practically) computable number. If $N$ would be this number, why should $N+1$ (larger than $N$) not be computable ? – Peter Jan 25 '23 at 07:14
  • This is what I expected. So to understand, there are no incalculable integers, regardless of how vast, they will still be essentially even, odd, prime, etc? The reason I am asking is because I am curious that for questions about infinite processes, like the Goldbach Conjecture, there shouldn't be any boundary. The question can be considered out to infinity without restriction other than our physical resources. Attempting to determine if a gargantuan number is the sum of two primes seems like it would involve some calculation so I wasn't sure if it would blow up some how. – Juel Herbranson Jan 25 '23 at 08:24
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    @Juel: well, there is a little bit of faith involved in believing that extremely large integers continue to behave exactly the same way as the familiar ones. This is more or less equivalent to believing that the principle of mathematical induction really holds. This is an extremely standard belief and once you hold it there's no issue with discussing, on a theoretical level at least, the primality of integers so large their descriptions don't fit in the universe, etc. etc. – Qiaochu Yuan Jan 25 '23 at 08:30
  • @QiaochuYuan This is really the center of my question. I can see a physical limit, but I've heard people discuss incalculable processes, etc. and I was interested if maybe the idea of certain numbers even lending themselves to the concept of encoding on any level is just not with in the realm of abstract consideration. I was surprised by the Godel result, so my mind has been wondering about what else may be limited by languages like math. And thank you for the discussion link. – Juel Herbranson Jan 25 '23 at 08:39
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    You might want to look at the problem of computing the "Busy Beaver" numbers (which are very very very large integers). – Gerry Myerson Jan 25 '23 at 08:57
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    Have you looked at the Busy Beaver numbers, Juel? – Gerry Myerson Jan 26 '23 at 21:50
  • Yes, I've only learned about it, I have not worked with it directly. That was one of the results that led me to Godel and now Goldbach. I was actually wondering if Goldback would become unwieldily in the same way as BB if the numbers considered were astronomically larger (astronomically is an understatement in this case). – Juel Herbranson Jan 27 '23 at 02:06
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    Well, I suppose if you asked whether twice the $n$th Busy Beaver number satisfies the Goldbach conjecture then that would become "unwieldy", but that would be entirely down to the Busy Beaver, and not tell you anything interesting about Goldbach. – Gerry Myerson Jan 27 '23 at 22:58

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Every integer is computable in the technical sense, because an integer is ultimately just some finite string of digits and so there always exists a program whose code simply consists of "print (string of digits)."

If the integer is extremely large, then this program will also be extremely large, and this is mostly unavoidable by a straightforward counting argument which establishes that most strings have the largest possible Kolmogorov complexity. But ignoring the lack of the necessary resources to physically write down such extremely large programs, this isn't an issue for the technical definition of a computable number.

The discussion here may also be helpful.

Qiaochu Yuan
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