It seems that most misundestood my post. Most only thought about the double spending problem in the domain of bitcoin. But my question is more general. The double-spending problem is not limited to bitcoin, but most people here only think about bitcoin when talking about double spending problem.

That is not correct, as the double spending problem is a more general one. Bitcoin has one possible solution to the double-spending problem, which is trusting the collective party which controls the majority of the network's hashing power.

As stated, there are many possible solutions to the double spending problem. Other solutions than that of bitcoin usually involve a centralized trusted party, e.g. linden dollars or the taken down liberty reserve. But all of the known solutions involve trust in some way. My question now was not about bitcoin. It was about the more general double spending problem and if there exists a mathematical proof that the double spending problem can only be solved with trust in one or the other way, or if this proof does not exist, if it might be possible to find a solution which does not involve trust.

Yes, I'm very aware of all the trivial prose intuitive explanations why trust is required. But prose explanations can mislead, while mathematical proof is guaranteed to be correct (provided there is no mistake in proof and premises).

I'm aware of them so you don't need to repeat them here, please! And I know bitcoin's solution to it extremely well, so please also don't explain bitcoin's individual solution here any more. I know it.

Finally, if you are not a maths pro, don't comment here.