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There exist numbers like Chaitin's constant which can be defined in natural language but for which it can be proven that there exists no finite algorithm to compute it to arbitrary precision, can a number exist is there any famous example of a number which is definable using natural language, and which can be computed to arbitrary precision by a finite algorithm, but can nevertheless not be computed by us because its impossible to find this algorithm, or prove that a given algorithm computes this number or not?

(This appears to be somewhat similar to How do uncomputable numbers relate to uncomputable functions?)

(Please point out if this question is too ambiguous or more suitable for a different forum)

codebender
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    Of course there are "tricky" answers. Let our number be $1$ if Goldbach's conjecture is true, and $0$ otherwise. Then there is an algorithm that computes this number: indeed, there are two simple algorithms I can think of, and one of those computes the number. But (of course) we do not know which of those two algorithms it is. – GEdgar May 05 '24 at 13:18

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Taken literally, your question is an oxymoron: it appears to contradict itself. "Computable" means "can be computed" so it is like asking: is there something red which is not red?

However, I guess that you want something less literal: a real number which is in principle computable but not in practice. Such numbers fairly certainly exist but any example could cease to be one tomorrow as someone might figure out how to compute it.

Chaitin's Constant is interesting because it is based on a problem that is provably impossible: The Halting Problem. So, it will never be computable even in principle.

To get what you want, pick your favourite conjecture which is believed to be provably true or false but has not been proved, disproved, or proven to be undecidable . In a comment, GEdgar gives the Goldbach conjecture. The Riemann Hypothesis is another one. Now, you can define your number, similarly to Chaitin's Constant, based on that.

$$Codebender's \ Constant = \begin{cases} 0, & RH \ is \ false, \\ 1, & RH \ is \ true, \\ -1, & RH \ is \ undecidable \end{cases}$$

I see that there is speculation for both conjectures that they are undecidable so these might not be computable. I added the $-1$ to cover this case. That is a bit beyond me, see this discussion on Math Overflow if you are interested: Can the Riemann hypothesis be undecidable?

If someone solves the Riemann Hypothesis tomorrow then this ceases to be an example.

Riemann Hypothesis

badjohn
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    Rather than involving the unnecessary complications of the Riemann Hypothesis or the Goldbach conjecture, maybe use an example like like $C = 1$ if and only if there is intelligent life on Jupiter, or $C=1$ if there was a rain of frogs in Xi'an in May of 628. The point is that your example is not actually mathematical. Its non-computability comes from contingent facts about the world whose truth happens to be unknown and possibly unknowable. – MJD May 05 '24 at 15:17
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    Take Chaitin's constant, and truncate it to the first $1024^{1024^{1024^{1024}}}$ digits. Classical computability says this is computable, but 1) there are only $10^{80}$ baryons in the observable universe, and the number of bits to just store the answer is unimaginably more than that and 2) no one will ever know that many bits anyway, because this should be big enough to contain diagonal machines for every algorithm we could practically run to check (if not, add more exponents). – Dan Doel May 05 '24 at 15:24
  • @MJD An interesting idea but then we get into a debate about what constitutes intelligent life. I guessed that the OP wanted a purely mathematical answer but I will let them decide whether they prefer this idea. I am not sure what you mean by my example not being mathematical; plenty of mathematicians work on the status of the Riemann Hypothesis. – badjohn May 05 '24 at 15:25
  • @DanDoel That is a good idea. – badjohn May 05 '24 at 15:26