Pub Date : 2024-08-08DOI: 10.1007/s00153-024-00940-7
Marina Dorzhieva, Rodney Downey, Ellen Hammatt, Alexander G. Melnikov, K. Ng
{"title":"Punctually presented structures II: comparing presentations","authors":"Marina Dorzhieva, Rodney Downey, Ellen Hammatt, Alexander G. Melnikov, K. Ng","doi":"10.1007/s00153-024-00940-7","DOIUrl":"https://doi.org/10.1007/s00153-024-00940-7","url":null,"abstract":"","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141928836","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-06-08DOI: 10.1007/s00153-024-00927-4
M. Peretyat’kin
{"title":"The Tarski–Lindenbaum algebra of the class of strongly constructivizable models with $$omega $$-stable theories","authors":"M. Peretyat’kin","doi":"10.1007/s00153-024-00927-4","DOIUrl":"https://doi.org/10.1007/s00153-024-00927-4","url":null,"abstract":"","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2024-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141368907","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-21DOI: 10.1007/s00153-024-00931-8
Paul Howard
{"title":"Separablilty of metric measure spaces and choice axioms","authors":"Paul Howard","doi":"10.1007/s00153-024-00931-8","DOIUrl":"https://doi.org/10.1007/s00153-024-00931-8","url":null,"abstract":"","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2024-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141117609","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-20DOI: 10.1007/s00153-024-00929-2
Konstantin Kovalyov
{"title":"Fragments of IOpen","authors":"Konstantin Kovalyov","doi":"10.1007/s00153-024-00929-2","DOIUrl":"https://doi.org/10.1007/s00153-024-00929-2","url":null,"abstract":"","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2024-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141120436","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-01-01Epub Date: 2023-08-11DOI: 10.1007/s00153-023-00888-0
Damian Sobota, Lyubomyr Zdomskyy
We prove that if is an infinite Boolean algebra in the ground model V and is a notion of forcing adding any of the following reals: a Cohen real, an unsplit real, or a random real, then, in any -generic extension V[G], has neither the Nikodym property nor the Grothendieck property. A similar result is also proved for a dominating real and the Nikodym property.
我们证明,如果 A 是基础模型 V 中的一个无穷布尔代数,而 P 是一个强制添加以下任何一个实数的概念:一个科恩实数、一个未分割实数或一个随机实数,那么在任何 P 代扩展 V[G] 中,A 既不具有尼科德姆性质,也不具有格罗thendieck 性质。对于支配实数和尼科戴姆性质,也证明了类似的结果。
{"title":"Convergence of measures after adding a real.","authors":"Damian Sobota, Lyubomyr Zdomskyy","doi":"10.1007/s00153-023-00888-0","DOIUrl":"10.1007/s00153-023-00888-0","url":null,"abstract":"<p><p>We prove that if <math><mi>A</mi></math> is an infinite Boolean algebra in the ground model <i>V</i> and <math><mi>P</mi></math> is a notion of forcing adding any of the following reals: a Cohen real, an unsplit real, or a random real, then, in any <math><mi>P</mi></math>-generic extension <i>V</i>[<i>G</i>], <math><mi>A</mi></math> has neither the Nikodym property nor the Grothendieck property. A similar result is also proved for a dominating real and the Nikodym property.</p>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10787011/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"52099468","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-08-12DOI: 10.1007/s00153-023-00889-z
Andrés Cordón-Franco, F. F. Lara-Martín
{"title":"Semi-honest subrecursive degrees and the collection rule in arithmetic","authors":"Andrés Cordón-Franco, F. F. Lara-Martín","doi":"10.1007/s00153-023-00889-z","DOIUrl":"https://doi.org/10.1007/s00153-023-00889-z","url":null,"abstract":"","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2023-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44525200","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-08-04DOI: 10.1007/s00153-023-00887-1
Omer Ben-Neria
{"title":"A Mathias criterion for the Magidor iteration of Prikry forcings","authors":"Omer Ben-Neria","doi":"10.1007/s00153-023-00887-1","DOIUrl":"https://doi.org/10.1007/s00153-023-00887-1","url":null,"abstract":"","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2023-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41471612","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-07-29DOI: 10.1007/s00153-023-00877-3
Kenji Miyamoto, G. Moser
{"title":"Herbrand complexity and the epsilon calculus with equality","authors":"Kenji Miyamoto, G. Moser","doi":"10.1007/s00153-023-00877-3","DOIUrl":"https://doi.org/10.1007/s00153-023-00877-3","url":null,"abstract":"","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2023-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44538500","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-06-13DOI: 10.1007/s00153-023-00871-9
Paul Howard, Eleftherios Tachtsis
We provide a general criterion for Fraenkel–Mostowski models of ({textsf{ZFA}}) (i.e. Zermelo–Fraenkel set theory weakened to permit the existence of atoms) which implies “every linearly ordered set can be well ordered” (({textsf{LW}})), and look at six models for ({textsf{ZFA}}) which satisfy this criterion (and thus ({textsf{LW}}) is true in these models) and “every Dedekind finite set is finite” (({textsf{DF}}={textsf{F}})) is true, and also consider various forms of choice for well-ordered families of well orderable sets in these models. In Model 1, the axiom of multiple choice for countably infinite families of countably infinite sets (({textsf{MC}}_{aleph _{0}}^{aleph _{0}})) is false. It was the open question of whether or not such a model exists (from Howard and Tachtsis “On metrizability and compactness of certain products without the Axiom of Choice”) that provided the motivation for this paper. In Model 2, which is constructed by first choosing an uncountable regular cardinal in the ground model, a strong form of Dependent choice is true, while the axiom of choice for well-ordered families of finite sets (({textsf{AC}}^{{textsf{WO}}}_{{textsf{fin}}})) is false. Also in this model the axiom of multiple choice for well-ordered families of well orderable sets fails. Model 3 is similar to Model 2 except for the status of ({textsf{AC}}^{{textsf{WO}}}_{{textsf{fin}}}) which is unknown. Models 4 and 5 are variations of Model 3. In Model 4 ({textsf{AC}}_{textrm{fin}}^{{textsf{WO}}}) is true. The construction of Model 5 begins by choosing a regular successor cardinal in the ground model. Model 6 is the only one in which (2{mathfrak {m}} = {mathfrak {m}}) for every infinite cardinal number ({mathfrak {m}}). We show that the union of a well-ordered family of well orderable sets is well orderable in Model 6 and that the axiom of multiple countable choice is false.
{"title":"Models of ({{textsf{ZFA}}}) in which every linearly ordered set can be well ordered","authors":"Paul Howard, Eleftherios Tachtsis","doi":"10.1007/s00153-023-00871-9","DOIUrl":"10.1007/s00153-023-00871-9","url":null,"abstract":"<div><p>We provide a general criterion for Fraenkel–Mostowski models of <span>({textsf{ZFA}})</span> (i.e. Zermelo–Fraenkel set theory weakened to permit the existence of atoms) which implies “every linearly ordered set can be well ordered” (<span>({textsf{LW}})</span>), and look at six models for <span>({textsf{ZFA}})</span> which satisfy this criterion (and thus <span>({textsf{LW}})</span> is true in these models) and “every Dedekind finite set is finite” (<span>({textsf{DF}}={textsf{F}})</span>) is true, and also consider various forms of choice for well-ordered families of well orderable sets in these models. In Model 1, the axiom of multiple choice for countably infinite families of countably infinite sets (<span>({textsf{MC}}_{aleph _{0}}^{aleph _{0}})</span>) is false. It was the open question of whether or not such a model exists (from Howard and Tachtsis “On metrizability and compactness of certain products without the Axiom of Choice”) that provided the motivation for this paper. In Model 2, which is constructed by first choosing an uncountable regular cardinal in the ground model, a strong form of Dependent choice is true, while the axiom of choice for well-ordered families of finite sets (<span>({textsf{AC}}^{{textsf{WO}}}_{{textsf{fin}}})</span>) is false. Also in this model the axiom of multiple choice for well-ordered families of well orderable sets fails. Model 3 is similar to Model 2 except for the status of <span>({textsf{AC}}^{{textsf{WO}}}_{{textsf{fin}}})</span> which is unknown. Models 4 and 5 are variations of Model 3. In Model 4 <span>({textsf{AC}}_{textrm{fin}}^{{textsf{WO}}})</span> is true. The construction of Model 5 begins by choosing a regular successor cardinal in the ground model. Model 6 is the only one in which <span>(2{mathfrak {m}} = {mathfrak {m}})</span> for every infinite cardinal number <span>({mathfrak {m}})</span>. We show that the union of a well-ordered family of well orderable sets is well orderable in Model 6 and that the axiom of multiple countable choice is false.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2023-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50023467","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-06-13DOI: 10.1007/s00153-023-00871-9
Paul Howard, E. Tachtsis
{"title":"Models of ZFAdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$${{textsf{ZFA}}}$$end{document} in which every linearly ","authors":"Paul Howard, E. Tachtsis","doi":"10.1007/s00153-023-00871-9","DOIUrl":"https://doi.org/10.1007/s00153-023-00871-9","url":null,"abstract":"","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2023-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42828924","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}