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.

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.

- 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:

- $\emptyset \in \omega$
- If $a\in \omega$ then $a\cup \{a\}\in \omega.$

| | | | | created: 2019-03-08 13:59:51 | modified: 2020-07-12 09:51:50 | by: *bookofproofs* | references: [656], [8055]

[8055] **Hoffmann, D.**: “Forcing, Eine Einführung in die Mathematik der Unabhängigkeitsbeweise”, Hoffmann, D., 2018

[656] **Hoffmann, Dirk W.**: “Grenzen der Mathematik – Eine Reise durch die Kerngebiete der mathematischen Logik”, Spektrum Akademischer Verlag, 2011