The Hartogs-Lindenbaum Spectrum of Symmetric Extensions

TL;DR

This paper generalizes a classic set theory equivalence, introduces the Hartogs-Lindenbaum spectrum concept, and analyzes its structure in models obtained via symmetric extensions, revealing a rigid pattern.

Contribution

It introduces the Hartogs-Lindenbaum spectrum and characterizes its structure in symmetric extension models of ZFC, extending classical equivalences in set theory.

Findings

All spectra in symmetric extension models follow a rigid pattern.

Extended the equivalence of $ ext{AC}_{ ext{WO}}$ to related statements.

Provided a detailed analysis of spectra structure in models of ZF.

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.