Let me write down my train of thoughts to make my question clear.
If the collection of elements $\{0,1,2,3,....\}$ is defined as the set $\mathbb{N}$. Then if $\omega$ is the first transfinite ordinal. Is there a corresponding definition (set) for the following collection of elements:
$\{\omega, \omega +1, \omega +2, ...\}$
And/or
$\{0,1,2,...,\omega, \omega +1, \omega +2, ...\}$
And/or
$\{\omega, \omega +1, \omega +2, ..., \omega^2, \omega^2 +1, \omega^2 +2, ..., \omega^3, \omega^3 +1, \omega^3 +2, ..., \omega^{\omega}\}$
And/or
$\{0,1,2,..., \omega, \omega +1, \omega +2, ..., \omega^2, \omega^2 +1, \omega^2 +2, ..., \omega^3, \omega^3 +1, \omega^3 +2, ..., \omega^{\omega}\}$ ?
P.S. If you can write down your sources I would deeply appreciate it
In the usual development of set theory, each ordinal is represented as the set of all smaller ordinals. (In particular, this makes $\omega$ a name for the set more elementarily known as $\mathbb N$). Therefore, the set $$\{0,1,2,...,\omega, \omega +1, \omega +2, ...\}$$ is the ordinal $\omega\cdot 2$, also known as $\omega+\omega$.
The best notation for $$ \{\omega, \omega +1, \omega +2, ...\}$$ would be $\omega\cdot 2 \setminus \omega$.
The set of all ordinals up to and including $\omega^\omega$ is $\omega^\omega+1$.