I know that in general ordinal addition is not strictly increasing in the left argument (as in $0+ \omega = n+\omega$).
Now I have a fixed countable ordinal $\delta$ and a natural number $k$. My intuition is that the set $\{ \omega^k \eta+\delta \mid \eta \text{ countable} \}$ contains uncountably many elements because there are uncountably many $\eta$ and most of them are much bigger than $\delta$ whence for most $\eta_1\neq \eta_2$ adding $\delta$ should preserve the inequality.
But I cannot come up with a proof for my intuition. Can anyone help me?
Let $X$ denote your set. Let $\alpha$ be the smallest ordinal bigger than $X$. If $X$ is countable then $\alpha$ is countable union of countable sets, so it is countable. But then $\omega^k\alpha+\delta >\alpha$. Contradiction.