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) is a transitive set $(X,\in_X)$ which is strictly-ordered and well-ordered with respect to the contained relation $\in_X.$
Table of Contents
- Proposition: Equivalent Notions of Ordinals
- Proposition: Ordinals Are Downward Closed
- Lemma: Equivalence of Set Inclusion and Element Inclusion of Ordinals
- Theorem: Trichotomy of Ordinals
Definitions: 1 2 3
Lemmas: 4 5 6 7
Motivations: 8 9
Proofs: 12 13 14 15 16 17 18
Propositions: 19 20 21
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