Order type omega

Omega is the smallest infinite ordinal number. It can be constructed as an ordinal by taking the set of all finite ordinals (all ordinals with a cardinality of less than aleph null), ordered under the element of relation.

It is the order type of the set of natural numbers.

Limit ordinal
Since omega is a limit ordinal, it can be written as the union of an infinite series of ordinals. One of these, with cardinality aleph null (equal to the cofinality of omega), is $$0 \cup 1 \cup 2 \cup 3 \cdot\cdot\cdot$$, including all finite ordinals.