Arrow's impossibility theorem comes to mind. To quote Wikipedia:
In short, the theorem states that no rank-order electoral system can be designed that always satisfies these three "fairness" criteria:
- If every voter prefers alternative X over alternative Y, then the group prefers X over Y.
- If every voter's preference between X and Y remains unchanged, then the group's preference between X and Y will also remain unchanged (even if voters' preferences between other pairs like X and Z, Y and Z, or Z and W change).
- There is no "dictator": no single voter possesses the power to always determine the group's preference.
More in the spirit of the question: The set of fair rank-ordered electoral systems is empty.