Burali-Forti Paradox

Theorem

The existence of the set of all ordinals leads to a contradiction.

Proof

Suppose that the collection of all ordinals is a set.

Let this set be denoted as $\On$.

From the corollary of ordinals are well-ordered, it is seen that $\Epsilon {\restriction_{\On} }$ is a strict well-ordering on $\On$.

By Element of Ordinal is Ordinal, it is seen that $\On$ is transitive.

And so $\On$ is itself an ordinal.

Since $\On$ is an ordinal, it follows from hypothesis that $\On \in \On$.

Since $\On$ is an ordinal, it follows from Ordinal is not Element of Itself that $\On \notin \On$.

But this is a contradiction.

Therefore a paradox has been formed.

$\blacksquare$

Also see

• The Ordinal Class, $\On$
• Russell's Paradox, another paradox in naive set theory.

Source of Name

This entry was named for Cesare Burali-Forti.

Sources

• 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 7.13$