Motivation: Burali-Forti Paradox
(related to Part: Ordinal Numbers)
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.
- Hoffmann, Dirk W.: "Grenzen der Mathematik - Eine Reise durch die Kerngebiete der mathematischen Logik", Spektrum Akademischer Verlag, 2011