Burali-Forti Paradox

Paradox
The ordinal of the set of all ordinals must be larger than any of the ordinals in that set.

Hence this would be an ordinal which is not contained in that set.

That is, it is an ordinal which is not contained in the set of all ordinals.

Hence the set of all ordinals is not an allowable set.

Refutation
This paradox is an antinomy caused by the incorrect assumption that one can create a set of all ordinals.

This is formally expressed in Existence of Set of Ordinals leads to Contradiction.

Also known as
Sometimes presented with an apostrophe: Burali-Forti's Paradox

Also see

 * Russell's Paradox, another paradox in naive set theory.