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. 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

Definitions: 1 2 3
Lemmas: 4 5 6 7
Motivations: 8 9
Parts: 10
Persons: 11
Proofs: 12 13 14 15 16 17 18
Propositions: 19 20 21
Theorems: 22

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  1. Hoffmann, Dirk W.: "Grenzen der Mathematik - Eine Reise durch die Kerngebiete der mathematischen Logik", Spektrum Akademischer Verlag, 2011
  2. Hoffmann, D.: "Forcing, Eine Einführung in die Mathematik der Unabhängigkeitsbeweise", Hoffmann, D., 2018


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