Motivation: BuraliForti Paradox
(related to Part: Ordinal Numbers)
Historically, the Italian Cesare BuraliForti (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 socalled BuraliForti paradox remained a paradox until the set theory got a better axiomatic foundation in the form of the ZermeloFraenkel axioms. Since then, the BuraliForti 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 ZermeloFraenkel 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 ZermeloFraenkel axiom, when the Cantor's principle of comprehension was replaced by the more cautious axiom of separation.
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References
Bibliography
 Hoffmann, Dirk W.: "Grenzen der Mathematik  Eine Reise durch die Kerngebiete der mathematischen Logik", Spektrum Akademischer Verlag, 2011