# 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.