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}