Burali-Forti Paradox
Jump to navigation
Jump to search
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: $\S 7.13$