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The corollary to the Mostowski’s Theorem provides a possibility to create a transitive set $(X,\in_X)$ which is ordered with respect to the contained relation $\in_X$ in exactly the same way as any given strictly-ordered, well-ordered set.

This leads to a possibility for choosing transitive sets, which are well-ordered with respect to the contained relation $\in_X$ as standard representatives of all strictly-ordered, well-ordered sets. This is the concept ordinals or ordinal numbers.

Definition: Ordinal Number

An ordinal (or ordinal number)1 is a transitive set $(X,\in_X)$ which is strictly-ordered and well-ordered with respect to the contained relation $\in_X.$

1 Please note that ordinal numbers are not “numbers” in the traditional sense, but sets.

| | | | | created: 2014-06-28 21:28:41 | modified: 2020-06-20 06:23:00 | by: bookofproofs | references: [656], [8055]

1.Proposition: Equivalent Notions of Ordinals

2.Proposition: Ordinals Are Downward Closed

3.Lemma: Equivalence of Set Inclusion and Element Inclusion of Ordinals

4.Theorem: Trichotomy of Ordinals

Edit or AddNotationAxiomatic Method

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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