Burali-Forti paradox
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The Burali-Forti paradox demonstrates that the ordinal numbers, unlike the natural numbers, do not form a set. The ordinal numbers can be defined as the class consisting of all sets x on which set inclusion is a total order and each element of x is also a subset of x.

E.g.,

0 is defined as {}, the empty set

1 is defined as {0} which can be written as

2 is defined as {0, 1} which can be written as }

3 is defined as {0, 1, 2} which can be written as , }}

...

in general, n is defined as {0, 1, 2, ... n−1}

So all natural numbers are ordinal numbers, and the set of natural numbers is an ordinal number itself. By this definition, if the ordinal numbers formed a set, that set would then be an ordinal number greater than any number in the set. This contradicts the assertion that the set contains all ordinal numbers.

The original article can be found at: http://en.wikipedia.org/wiki/Burali-Forti_paradox.
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