Forcing axioms and the complexity of non-stationary ideals

We study the influence of strong forcing axioms on the complexity of the non-stationary ideal on $\omega_2$ and its restrictions to certain cofinalities. Our main result shows that the strengthening $\mathrm{MM}^{++}$of Martin's Maximum does not decide whether the restriction of the non-station...

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Detalles Bibliográficos
Autores: Cox, Sean, Lücke, Philipp
Tipo de recurso: artículo
Estado:Versión publicada
Fecha de publicación:2022
País:España
Institución:Universidad de Barcelona
Repositorio:Dipòsit Digital de la UB
OAI Identifier:oai:diposit.ub.edu:2445/217445
Acceso en línea:https://hdl.handle.net/2445/217445
Access Level:acceso abierto
Palabra clave:Teoria de conjunts
Lògica matemàtica
Set theory
Mathematical logic
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spelling Forcing axioms and the complexity of non-stationary idealsCox, SeanLücke, PhilippTeoria de conjuntsLògica matemàticaSet theoryMathematical logicWe study the influence of strong forcing axioms on the complexity of the non-stationary ideal on $\omega_2$ and its restrictions to certain cofinalities. Our main result shows that the strengthening $\mathrm{MM}^{++}$of Martin's Maximum does not decide whether the restriction of the non-stationary ideal on $\omega_2$ to sets of ordinals of countable cofinality is $\Delta_1$-definable by formulas with parameters in $\mathrm{H}\left(\omega_3\right)$. The techniques developed in the proof of this result also allow us to prove analogous results for the full non-stationary ideal on $\omega_2$ and strong forcing axioms that are compatible with CH. Finally, we answer a question of S . Friedman, Wu and Zdomskyy by showing that the $\Delta_1$-definability of the non-stationary ideal on $\omega_2$ is compatible with arbitrary large values of the continuum function at $\omega_2$.Springer Verlag2022info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionapplication/pdfhttps://hdl.handle.net/2445/217445Articles publicats en revistes (Matemàtiques i Informàtica)reponame:Dipòsit Digital de la UBinstname:Universidad de BarcelonaInglésReproducció del document publicat a: https://doi.org/10.1007/s00605-022-01734-wMonatshefte für Mathematik, 2022, vol. 199, num.1, p. 45-84https://doi.org/10.1007/s00605-022-01734-wcc by (c) Sean cox et al., 2022http://creativecommons.org/licenses/by/3.0/es/info:eu-repo/semantics/openAccessoai:diposit.ub.edu:2445/2174452026-05-27T06:46:51Z
dc.title.none.fl_str_mv Forcing axioms and the complexity of non-stationary ideals
title Forcing axioms and the complexity of non-stationary ideals
spellingShingle Forcing axioms and the complexity of non-stationary ideals
Cox, Sean
Teoria de conjunts
Lògica matemàtica
Set theory
Mathematical logic
title_short Forcing axioms and the complexity of non-stationary ideals
title_full Forcing axioms and the complexity of non-stationary ideals
title_fullStr Forcing axioms and the complexity of non-stationary ideals
title_full_unstemmed Forcing axioms and the complexity of non-stationary ideals
title_sort Forcing axioms and the complexity of non-stationary ideals
dc.creator.none.fl_str_mv Cox, Sean
Lücke, Philipp
author Cox, Sean
author_facet Cox, Sean
Lücke, Philipp
author_role author
author2 Lücke, Philipp
author2_role author
dc.subject.none.fl_str_mv Teoria de conjunts
Lògica matemàtica
Set theory
Mathematical logic
topic Teoria de conjunts
Lògica matemàtica
Set theory
Mathematical logic
description We study the influence of strong forcing axioms on the complexity of the non-stationary ideal on $\omega_2$ and its restrictions to certain cofinalities. Our main result shows that the strengthening $\mathrm{MM}^{++}$of Martin's Maximum does not decide whether the restriction of the non-stationary ideal on $\omega_2$ to sets of ordinals of countable cofinality is $\Delta_1$-definable by formulas with parameters in $\mathrm{H}\left(\omega_3\right)$. The techniques developed in the proof of this result also allow us to prove analogous results for the full non-stationary ideal on $\omega_2$ and strong forcing axioms that are compatible with CH. Finally, we answer a question of S . Friedman, Wu and Zdomskyy by showing that the $\Delta_1$-definability of the non-stationary ideal on $\omega_2$ is compatible with arbitrary large values of the continuum function at $\omega_2$.
publishDate 2022
dc.date.none.fl_str_mv 2022
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv https://hdl.handle.net/2445/217445
url https://hdl.handle.net/2445/217445
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv Reproducció del document publicat a: https://doi.org/10.1007/s00605-022-01734-w
Monatshefte für Mathematik, 2022, vol. 199, num.1, p. 45-84
https://doi.org/10.1007/s00605-022-01734-w
dc.rights.none.fl_str_mv cc by (c) Sean cox et al., 2022
http://creativecommons.org/licenses/by/3.0/es/
info:eu-repo/semantics/openAccess
rights_invalid_str_mv cc by (c) Sean cox et al., 2022
http://creativecommons.org/licenses/by/3.0/es/
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.publisher.none.fl_str_mv Springer Verlag
publisher.none.fl_str_mv Springer Verlag
dc.source.none.fl_str_mv Articles publicats en revistes (Matemàtiques i Informàtica)
reponame:Dipòsit Digital de la UB
instname:Universidad de Barcelona
instname_str Universidad de Barcelona
reponame_str Dipòsit Digital de la UB
collection Dipòsit Digital de la UB
repository.name.fl_str_mv
repository.mail.fl_str_mv
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