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.
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 $\in_X.$
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
Please note that ordinal numbers are not "numbers" in the traditional sense, but sets. ↩