Kleinberg sequences and partition cardinals below \(\pmb\delta_5^1\) (Q2773311)

For a~cardinal~\(\kappa\) with the strong partition property and a~normal ultrafilter~\(\mu\) on~\(\kappa\), the sequence of cardinals \(\kappa_n\), \(n\geq 1\), defined by \(\kappa_1=\kappa\) and \(\kappa_{n+1}=(\kappa_n)^\kappa/\mu\), is called the Kleinberg sequence derived from~\(\mu\). If \(\lambda<\kappa\) is regular and \(\kappa\) is a~cardinal with the strong partition property then the filter~\(\mathcal C^\lambda_\kappa\) generated by \(\lambda\)-closed unbounded sets in~\(\kappa\) is a~normal ultrafilter on~\(\kappa\). Under AD, by a~result of S.~Jackson, the projective ordinal~\(\pmb\delta^1_3\) is a~cardinal with the strong partition property. Assuming~AD, the author considers the three normal filters on~\(\pmb\delta^1_3\), \(\mu_0=\mathcal C^\omega_{\pmb\delta^1_3}\), \(\mu_1=\mathcal C^\omega_{\pmb\delta^1_3}\), \(\mu_2=\mathcal C^\omega_{\pmb\delta^1_3}\), and proves that \(\kappa_n^{\mu_0}=\aleph_{\omega+n}\), \(\kappa_n^{\mu_1}=\aleph_{\omega\cdot n+1}\), and \(\kappa_n^{\mu_2}= \aleph_{\omega+\omega^\omega\cdot(n-1)+1}\).