Order type omega omega

Omega omega is the order type of the ordinal number that contains all ordinals with a cardinality less than aleph omega, ordered according to the element of relation.

Limit ordinal
As a limit ordinal, it can be expressed as the union of a sequence of smaller ordinals. One of these, with a cardinality of aleph null (equal to the cofinality of omega omega), is $$\omega \cup \omega_1 \cup \omega_2 \cup \omega_3 \cdot\cdot\cdot$$.