The $\lambda^{\aleph_0}$ are cardinals of uncountable cofinality which are $\geq 2^{\aleph_0}$. But are they all the cardinals of uncountable cofinality which are $\geq 2^{\aleph_0}$? I think this need not hold: if $2^{\aleph_0} = \aleph_1$ but $\aleph_\omega^{\aleph_0} = \aleph_{\omega+2}$ then $\aleph_{\omega+1}$ is not a $\lambda^{\aleph_0}$ despite being of uncountable cofinality and $\geq 2^{\aleph_0}$; but this requires violating the Singular Cardinal Hypothesis, which is more difficult than Easton's theorem. I'm not sure, though. Maybe open a new question?
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