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.
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: , 
 Hoffmann, D.: “Forcing, Eine Einführung in die Mathematik der Unabhängigkeitsbeweise”, Hoffmann, D., 2018
 Hoffmann, Dirk W.: “Grenzen der Mathematik – Eine Reise durch die Kerngebiete der mathematischen Logik”, Spektrum Akademischer Verlag, 2011