The Burali-Forti paradox demonstrates that the class of all ordinals is not a set. If there were a set of all ordinals, $Ord$ , then it would follow that $Ord$ was itself an ordinal, and therefore that $Ord\in Ord$ . Even if sets in general are allowed to contain themselves, ordinals cannot since they are defined so that $\in$ is well founded over them.

This paradox is similar to both Russell's paradox and Cantor's paradox, although it predates both. All of these paradoxes prove that a certain object is ``too large'' to be a set.