User blog comment:Alemagno12/Some more set theory questions/@comment-1605058-20180105165055/@comment-1605058-20180108163437

Thanks, I see my mistake now - if \(\alpha\) is regular, it's not the case that \(\aleph_\alpha\) has cofinality greater than all smaller cardinals, because of the smaller successor cardinals. For example, for \(\alpha=\omega_1\), \(\aleph_1\) has the same cofinality.