I'm afraid I don't know how to post links.

I think there is already a main-page article on this subject, but the general idea is that Arrow's theorem assumes the voting system is preferential (you vote by ranking voters) and so you can get around it with a non-preferential system.

Range voting (each voter gives each candidate as score out of ten, and the candidate with the highest total wins) is the one that springs most easily to mind, but it has problems of its own, so somebody who knows more about the subject can probably give you a better example.

As for the main point, I doubt you actually put 100% confidence in either idea. In the unlikely event that either approach led you to a contradiction, would you just curl up in a ball and go insane, or abandon it.