Is it consistent with ZFC that $2^{\aleph_0} = \aleph_{2^{\aleph_0}}$?
2026-04-12 15:11:33.1776006693
Is it consistent with ZFC that $2^{\aleph_0}$ is a fixed point of the aleph function?
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Yes, it is consistent. The standard Cohen forcing allows you to set the continuum to anything with uncountable cofinality, and it is cardinal-preserving, so will preserve the property of being an aleph fixed point. So you can set it to any aleph fixed point that has uncountable cofinality, e.g. the $\omega_1$-st aleph fixed point.