Specifically what was proved was that the set of attributes which many people would agree are desirable from a preference-based voting system for a simple election (pick one candidate from a list) cannot all be achieved simultaneously for a single system.
This is called Arrow's impossibility theorem.
You can beat this if you're willing to change the possible criteria. For example, the "one person one vote" criterion might actually be unfair. We can recognise a decision by two wolves and one sheep on what to have for dinner is not made fair by giving each "person" a vote.