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Explanation: The Class of all Ordinals $\Omega$

We have seen that ordinal numbers are downward closed. But does it also hold for 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), $\Omega$ is not a set but a proper class, for which it is forbidden to be an element of a class (especially itself).

Nevertheless, all ordinals $\alpha\in\Omega$ build an infinite chain of being contained in each other. Moreover, 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:

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

This motivates the following lemma and definition of a successor of ordinal:

| | | | created: 2019-03-08 13:59:51 | modified: 2019-03-08 14:52:17 | by: bookofproofs | references: [656], [8055]

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Bibliography (further reading)

[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

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