Burali-Forti paradox

Burali-Forti paradox

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.