With von Neumann's definition of $\Omega$ , the ordinal set , we have $[0,\omega)=\omega \; (\forall \omega \in \Omega)$ . In fact with von Neumann approach , $\Omega$ cannot be considered as a set ( since $\Omega$ is itself well- ordered and belong to itself .

With the "modern" definition of $\Omega$ that consist in defining the ordinals as the equivalence classes of well-ordered sets ( two well ordered sets being equivalent if there exist a monotonous bijective mapping between them ) , do we still have this property ? Apparently yes because the demonstration by transfinite induction is still valid .