The Hartogs-Lindenbaum Spectrum of Symmetric Extensions

We expand the classic result that ${\mathsf{A}\mathsf{C}}_{\mathsf{W}\mathsf{O}}$ is equivalent to the statement “For all $X$, $\mathrm{\aleph }(X)={\mathrm{\aleph }}^{\ast }(X)$” by proving the equivalence of many more related statements. Then, we introduce the Hartogs-Lindenbaum spectrum of a model of $\mathsf{Z}\mathsf{F}$, and inspect the structure of these spectra in models that are obtained by a symmetric extension of a model of $\mathsf{Z}\mathsf{F}\mathsf{C}$. We prove that all such spectra fall into a very rigid pattern.