In the theory of transfinite ordinal numbers, 1. Every well ordered set has a unique ordinal number, 2. Every segment of ordinals (i.e., any set of ordinals arranged in natural order which contains all the predecessors of each of its elements) has an ordinal number which is greater than any ordinal in the segment, and 3. The set B of all ordinals in natural order is well ordered. Then by statements (3) and (1), B has an ordinal β. Since β is in B, it follows that β<β by (2), which is a contradiction.

ordinal number