= Singular cardinal enumeration
{title2=$\sigma_\alpha$}
= Increasing enumeration of uncountable singular cardinals
{synonym}
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 $\aleph_{\omega_1}$.
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