Full Idea
The Burali-Forti paradox says that if ordinals are defined by 'gathering' all their predecessors with the empty set, then is the set of all ordinals an ordinal? It is created the same way, so it should be a further member of this 'complete' set!
Clarification
Thus 'fourth' would be {empty, first, second, third}
Gist of Idea
The Burali-Forti paradox asks whether the set of all ordinals is itself an ordinal
Source
Michèle Friend (Introducing the Philosophy of Mathematics [2007], 2.3)
Book Reference
Friend,Michèle: 'Introducing the Philosophy of Mathematics' [Acumen 2007], p.26
A Reaction
This is an example (along with Russell's more famous paradox) of the problems that began to appear in set theory in the early twentieth century. See Idea 8675 for a modern solution.
Related Idea
Idea 8675 Paradoxes can be solved by talking more loosely of 'classes' instead of 'sets' [Friend]