By Cantor's normal form theorem, any ordinal $r$ can be expressed as: $r=\omega^{k_1}a_1 + \omega^{k_2}a_2 + \ldots$. ($k_1>k_2>\ldots$)
I want to know whether the class of all $a_i$'s is countable or not.
If it is not countable how do i prove a problem such as: $\omega^{k_1}a_1 + \omega^{k_2}a_2 + \omega^{k_3} + \ldots < \omega^{k_1}a_1 + \omega^{k_1}a_2 +\omega^{k_1}a_3 + \ldots$?