I have heard that $2^{\omega_1} = \omega_1^{\omega_1}$. Is that true? Why is that? I have tried to find a bijection between the set of all subsets of $\omega_1$ and the set of all functions $\alpha: \omega_1 \rightarrow \omega_1$ but I haven't found it. I have also tried to prove it with the definition with no success. Any help would be appreciated. Thank you ;)
2026-04-07 02:37:55.1775529475
Question about cardinals.
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Hint: $\omega_1^{\omega_1} \le (2^{\omega_1})^{\omega_1}$.