The Hartogs-Lindenbaum Spectrum of Symmetric Extensions

Abstract: 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.