The theorem stating that no ranked voting system can satisfy several fairness criteria with three or more options is Arrow's theorem.
More precisely, Arrow's impossibility theorem says that no social-welfare rule can simultaneously satisfy conditions including non-dictatorship, Pareto efficiency, and independence of irrelevant alternatives when voters may rank at least three alternatives in any order. Kenneth J. Arrow proved the result in the development of modern social choice theory; his 1951 book Social Choice and Individual Values presented the influential formulation.
The theorem does not say that every election is incoherent or that every voting method always fails. It identifies a trade-off among desirable principles. A classic illustration is a majority cycle: one group preference can place A above B, B above C, and C above A, even though every individual ranking is internally consistent.
A common mistake is applying Arrow's result to every possible voting system. Its standard framework concerns ordinal rankings and collective preference rules. Rated systems such as approval or score voting use additional information and are not covered in exactly the same way.