Proof That the Set of All Ordinal Numbers Does Not Exist

To prove that the set of all ordinal numbers does not exist, first prove that any set of ordinal numbers is well ordered. Suppose that a set A of ordinal numbers exists which is not well ordered. Then at least one subset B exists that does not have a least element. This set must contain a strictly decreasing sequence of ordinal numbers

This sequence is a subset of the setand this set is not well ordered because it contains a strictly decreasing infinite sequence. This is a contradiction sois well ordered.

Now suppose that a setof all ordinal numbers exists. By the argument above this set is well ordered. Letbe the ordinal number ofwith the orderingonmust be an element ofWe then have

This is a contradiction so there is no set of all ordinal numbers.