Is this known? Can we at least relate it to $\aleph_{2}$?
2026-03-30 12:02:45.1774872165
What is the value of $\aleph_{1}^{\aleph_{1}}$?
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As the comments already said:
$$2^{\aleph_1} \le \aleph_1^{\aleph_1} \le (2^{\aleph_1})^{\aleph_1} = 2^{\aleph_1}$$ so it equals $2^{\aleph_1} = |\mathscr{P}(\aleph_1)|$.
It's $\ge \aleph_2$, of course, but can be equal to it in some models of ZFC and strictly larger (say $\aleph_{100}$) in others. It's independent of ZFC (like CH is).