The Hartogs–Lindenbaum spectrum of symmetric extensions

We expand the classic result that ACWO$\mathsf {AC}_\mathsf {WO}$ is equivalent to the statement “For all X$X$, ℵ(X)=ℵ∗(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 ZF$\mathsf {ZF}$, and insp...

Full description

Saved in:
Bibliographic Details
Published inMathematical logic quarterly Vol. 70; no. 2; pp. 210 - 223
Main Author Ryan‐Smith, Calliope
Format Journal Article
LanguageEnglish
Published Berlin Wiley Subscription Services, Inc 01.05.2024
Subjects
Equivalence
Spectra
Online AccessGet full text

Cover

More Information
Summary:We expand the classic result that ACWO$\mathsf {AC}_\mathsf {WO}$ is equivalent to the statement “For all X$X$, ℵ(X)=ℵ∗(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 ZF$\mathsf {ZF}$, and inspect the structure of these spectra in models that are obtained by a symmetric extension of a model of ZFC$\mathsf {ZFC}$. We prove that all such spectra fall into a very rigid pattern.
Bibliography:ObjectType-Article-1
SourceType-Scholarly Journals-1
ObjectType-Feature-2
content type line 14
ISSN:0942-5616
1521-3870
DOI:10.1002/malq.202300047