Is there mathematical proof that the double-spending problem can be solved only by a trusted party, be it network majority, central trusted party, something in between like a trustweb, or even directly trusting the sender of the money? (Yes, if you trust the sender, there is no need for a third party.)

cf. http://en.wikipedia.org/wiki/Double-spending

From intuition, it is quite clear that without trust, the double-spending problem can't be solved. But intuition can often mislead. For that reason: Is there mathematical proof that trust is an implicit requirement for solving the double-spending problem?