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I know that $\omega_1$={$\alpha$ : $\alpha$ is a countable ordinal} is uncountable but what about the subset of $\omega_1$ of countable successor ordinals?

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Hint: There is an obvious bijection taking $\omega_1$ to the successor ordinals therein.

Hint 2: To be a successor ordinal is to be in the image of ...?

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Hint: Assume that it is countable; and take $\alpha=\sup\{\beta\mid\beta\text{ is a countable successor ordinal}\}$. Show that $\alpha$ is countable. Now, what can you say about $\alpha+1$? Conclude a contradiction.

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