◀ ▲ ▶Branches / Set-theory / Definition: The Class of all Ordinals `$\Omega$`
We have seen that ordinal numbers are downward closed. But does it also hold in the opposite direction? Is there a "set of all ordinals" $\Omega$?
The Burali-Forti paradox shows that such a set cannot exist. In other words, the notion of a set is "too narrow"
to be used for summing up all existing ordinal numbers.
Definition: The Class of all Ordinals $\Omega$
In the terms of the Neumann-Berneys-Gödel set theory (NBG), $\Omega$ as a collection of all ordinal numbers is not a set but a proper class.
Notes
- For a proper class, it is forbidden to be an element of a class (especially itself), therefore $\Omega\not\in\Omega.$
- Nevertheless, all ordinals $\alpha\in\Omega$ build an infinite chain of being contained in each other.
- Remember that we have already constructed the minimal inductive set $\omega$ independently from the discussion of ordinal numbers while we were talking about the Zermelo-Fraenkel axioms.
- $\omega$ can be visualized as follows:
Please note that $\omega\neq\Omega$, i.e. the set $\omega$ never can equal (!) the class $\Omega.$ However, it was constructed using a similar, recursive principle:
- $\emptyset \in \omega$
- $\emptyset \in \omega$
Mentioned in:
Proofs: 1
Propositions: 2
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References
Bibliography
- Hoffmann, Dirk W.: "Grenzen der Mathematik - Eine Reise durch die Kerngebiete der mathematischen Logik", Spektrum Akademischer Verlag, 2011
- Hoffmann, D.: "Forcing, Eine Einführung in die Mathematik der Unabhängigkeitsbeweise", Hoffmann, D., 2018
