Motivation: Burali-Forti Paradox

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 Russell'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.

Definitions: 1
Explanations: 2
Parts: 3
Persons: 4

Thank you to the contributors under CC BY-SA 4.0!

Hoffmann, Dirk W.: "Grenzen der Mathematik - Eine Reise durch die Kerngebiete der mathematischen Logik", Spektrum Akademischer Verlag, 2011