The Cantor Normal Form Theorem states that every ordinal $\alpha > 0$ can be uniquely expressed in the form $$\omega^{\beta_1}k_1 + \omega^{\beta_2}k_2 + \dots + \omega^{\beta_n}k_n$$ for some $n \ge 1$, positive integers $k_1,k_2,\dots,k_n$ and ordinals $\alpha \ge \beta_1 > \beta_2 > \dots > \beta_n$.

If I understand this correctly, then every countable ordinal's Cantor Normal Form is a finite-degree polynomial in $\omega$. My reasoning is that if $\beta$ is infinite, then $\omega^{\beta}$ is uncountable. This must be incorrect because it would imply that there are only countably many countable ordinals. Since $\aleph_1$ is the set of all countable ordinals and is uncountable, this is a contradiction.

My main question is where did I go wrong? If every countable set can't be written as a polynomial in $\omega$, what is a counter-example?

Thank you for your time.

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