Everybody knows about Mandelbrot set drawing computer programs. Program takes some point, builds sequence from it, and if found that sequence goes out of circle with 2 radius, then knows that this point does NOT belong to the set.
What about other points?
Sequences from them are infinite. Does this mean that we never can be 100% sure that these points belong to the set?
For example, if we take some point "A" (out of main cardioid)
can we ACTUALLY prove that this point belong to the set?
If not, then can this case be an example of Gödel's theorem, i.e. something true, but unprovable?
Thinking on this topic I thought that some points may be provable, like points inside main cardioid. Hence, there are possible provable theorems about other points too. There are probable numerous of them.
Hence, the incompleteness is escaping again: we can't be sure, that for some point "A" it will be never found a specific proof, that it belongs to the set. And if we have some point, for which the proof was not found for 1000 years, we still can't be sure that this point is unprovable...
Probably it is impossible to provide "problem domain" example of Gödel's theorem at all, because if it would be an example, it would be proven that it was unprovable :)