  We expand the classic result that $\mathsf{AC}_{\mathsf{WO}}$ is equivalent
to the statement "For all $X$, $\aleph(X)=\aleph^*(X)$" by proving the
equivalence of many more related statements. Then, we introduce the
Hartogs-Lindenbaum spectrum of a model of $\mathsf{ZF}$, and inspect the
structure of these spectra in models that are obtained by a symmetric extension
of a model of $\mathsf{ZFC}$. We prove that all such spectra fall into a very
rigid pattern.

Logic