Existence of Set of Ordinals leads to Contradiction
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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 Ordinals are Well-Ordered: Corollary, 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 by hypothesis that $\On \in \On$.
Since $\On$ is an ordinal, it follows from Ordinal is not Element of Itself that $\On \notin \On$.
This is a contradiction.
$\blacksquare$
Also known as
This theorem is sometimes presented as the Burali-Forti Paradox, but strictly speaking it is in fact a resolution to it.
Also see
- The class of all ordinals, $\On$
- Russell's Paradox, another paradox in naive set theory.
Sources
- 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 7.13$