Cantor's paradox

In mathematics, Cantor's paradox refers to two related paradoxes in naive set theory:

• The set of all cardinal numbers does not exist.
• The set of all sets does not exist.

In modern set theory, these are not paradoxes, as "all cardinal numbers" and "all sets" are proper classes, not sets.

These results are a consequence of Cantor's theorem, which states that the cardinality of a set is less than the cardinality of its power set (the set of all subsets).

Suppose that a set ${\displaystyle U}$ of all sets exists, and let ${\displaystyle T}$ be the union of all elements of ${\displaystyle U}$. Because it is the union, the cardinality of ${\displaystyle T}$ is greater than or equal to the cardinality of all elements of ${\displaystyle U}$.

Now let ${\displaystyle P(T)}$ be the power set of ${\displaystyle T}$. Since ${\displaystyle P(T)}$ is a set, it is an element of ${\displaystyle U}$. By Cantor's theorem, the cardinality of ${\displaystyle T}$ is strictly less than the cardinality of ${\displaystyle P(T)}$; this contradicts the cardinality of ${\displaystyle T}$ being the maximum of all cardinalities in ${\displaystyle U}$.

Thus, no set of all sets can exist.

The proof that the set of all cardinal numbers does not exist is similar. Since for each cardinal, there exists a set with that cardinality, a set of all cardinalities corresponds to a set of sets, one of each cardinality. From here, the proof is the same: the power set of the union gives a set that must be larger than every cardinality in the original set.