The Hartogs–Lindenbaum spectrum of symmetric extensions

Abstract

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