 Singular cardinal enumeration

Write ${\sigma }_{\alpha }$ for the $\alpha$th uncountable singular cardinal in increasing order. It begins with ${\sigma }_0={\aleph }_{\omega }$ and ${\sigma }_1={\aleph }_{\omega +\omega }$. At a nonzero limit ordinal index it is continuous exactly when the supremum of its earlier values is singular. It can jump when that supremum is a weakly inaccessible cardinal. The first member of uncountable cofinality occurs at index ${\omega }_1$, with value ${\aleph }_{{\omega }_1}$.