Is there a problem with an answer that can't be found?

Question

I have been thinking about something and I don't know whether it's possible or a contradiction, it is as follows:

Is there a mathematical problem for which we know there is an actual answer, but for which we also know that there is no way to ever calculate that answer?

I'm not talking about technical difficulties like really large numbers or anything, I'm talking about mathematical limitations.

So for instance calculating the number of atoms in the universe doesn't qualify. We know there is one exact answer at any given time and that we could find it (roughly) by doing a lot of calculations and observations. While it is in practice impossible to find an answer, it is not theoretically impossible and that is what I'm aiming at: theoretical impossibility.

Answer 1

Maybe this is what you are interested in, the so called Continuum Hypothesis which states that there is no set with a cardinality strict between the Natural Numbers, and the continuum.

Gödel and Cohen proved that this one can't be proved with ZF nor it can be disproved in ZF.

And Gödels Incompleteness theorems are a good example for mathematical limitations.

Answer 2

Look up Gödel's incompleteness theorems:

http://en.wikipedia.org/wiki/Gödel%27s_incompleteness_theorems

Answer 3

There are statements known (proved) to be true and also known not to have any proof in a specific formal proof system, as Gödel's incompleteness theorems show. There are things that are proven to exist (usually using the axiom of choice) while it can also be shown to be impossible to completely describe any example; non-measurable sets for the Lebesgue measure are a case in point. There are also infinite families of well defined yes/no questions for which we can show there is no computational method that will produce the right answer for all of them; the halting problem for Turing machines is an example.

However none of these seem to involve a question (one) for which "we know there is an actual answer". Rather in the former two cases we know about truth or existence in an higher abstract sense, while also knowing the absence of an "actual answer" (which I must interpret as something that in principle could be formulated), a concrete proof or construction that would exhibit that truth/existence. In the case of decidability, only infinite families of questions can provably undecidable; the problem for a single question is that either the oracle that says "yes" or the oracle that says "no" must have it right, so such a question is trivially decided by one of those oracles (though you are entitled to consider this solution cheating). So I would guess that your insistence that "we know there is no way to actually calculate that answer" more or less makes it impossible to really know that an "actual answer", taken in a strong sense, exists.

Answer 4

maybe Gregory Chaitin's $\Omega$ (see http://en.wikipedia.org/wiki/Chaitin%27s_constant ) is what you are searching for.

Quoting from Wikipedia, $\Omega$ is a real number that informally represents the probability that a randomly constructed program (given a specific encoding) will halt. The number exists, but it cannot be computed.