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 $X$, $\aleph (X)={\aleph }^{\ast }(X)$" 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.

arXiv comment16 pages; accepted version

References in corpus (2)

• The Bristol Model: an abyss called a Cohen real
• Which Pairs of Cardinals Can Be Hartogs and Lindenbaum Numbers of a Set?

Cited by in corpus (1)

• Intermediate models and Kinna--Wagner Principles