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arXiv · 2309.12100

The Hartogs-Lindenbaum Spectrum of Symmetric Extensions

Calliope Ryan-Smith
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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.

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Calliope Ryan-Smith. 2023-09-21. The Hartogs-Lindenbaum Spectrum of Symmetric Extensions. https://doi.org/10.1002/malq.202300047

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