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Historically, the Italian Cesare Burali-Forti (1861 – 1931) discovered that using the Cantor’s definition of a set, we could build the set $\Omega$ of all ordinal numbers. However, this set would lead to the following paradox:

• Since all elements $$\alpha\in\Omega$$ are ordinals, they are also transitive sets.
• Therefore it follows $$\alpha\subseteq \Omega$$.
• Since ordinals are downward closed, so $$\Omega$$ must be an ordinal itself.
• But then, both, $$\Omega\in\Omega$$ and $$\Omega=\Omega$$, which is a contradiction.

This so-called Burali-Forti paradox remained a paradox until the set theory got a better axiomatic foundation in the form of the Zermelo-Fraenkel axioms. Since then, the Burali-Forti paradox is only a paradox if we insist $\Omega$ to be a set. Today, $\Omega$ is no more considered a set, since it violates one of the Zermelo-Fraenkel axioms, the axiom of foundation.

Also, other paradoxes were discovered in the original Cantor’s set theory, including the Russel’s paradox, but those paradoxes were similarly removed by another Zermelo-Fraenkel axiom, when the Cantor’s principle of comprehension was replaced by the more cautious axiom of separation.

| | | | created: 2014-07-12 21:46:23 | modified: 2020-06-22 21:55:03 | by: bookofproofs | references: [656]