Possible values for \(2^{\aleph_n}\) and \(2^{\aleph_\omega}\)
Let \(m\) be a natural number with \(m>1\) and let \(\alpha: \omega\to \omega\) be a monotonic function with \(\alpha (n)>n\) for each \(n\). Assuming that \(\kappa\) is a \(P(\kappa)\)-hyper measurable cardinal, the authors build a forcing extension with \(2^{\aleph_n} =\aleph_{\alpha (n)}\) and \(2^{\aleph_\omega} =\aleph_{\omega+n}\). It was shown by Gitik and Magidor that for \(m>2\) this assumption is needed, whereas it can be shown that for \(m=2\) this assumption can be reduced.
scientific article; zbMATH DE number 1109893