Mathematical Logic Quarterly最新文献

{"title":"A proof-theoretic metatheorem for tracial von Neumann algebras","authors":"Liviu Păunescu,&nbsp;Andrei Sipoş","doi":"10.1002/malq.202200048","DOIUrl":"https://doi.org/10.1002/malq.202200048","url":null,"abstract":"<p>We adapt a continuous logic axiomatization of tracial von Neumann algebras due to Farah, Hart and Sherman in order to prove a metatheorem for this class of structures in the style of proof mining, a research programme that aims to obtain the hidden computational content of ordinary mathematical proofs using tools from proof theory.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"69 1","pages":"63-76"},"PeriodicalIF":0.3,"publicationDate":"2023-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50147575","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}

{"title":"Nice ℵ1 generated non-P-points, Part I","authors":"Saharon Shelah","doi":"10.1002/malq.202200070","DOIUrl":"https://doi.org/10.1002/malq.202200070","url":null,"abstract":"<p>We define a family of non-principal ultrafilters on <math>\u0000 <semantics>\u0000 <mi>N</mi>\u0000 <annotation>${mathbb {N}}$</annotation>\u0000 </semantics></math> which are, in a sense, very far from P-points. We prove the existence of such ultrafilters under reasonable conditions. In subsequent articles, we intend to prove that such ultrafilters may exist while no P-point exists. Though our primary motivations came from forcing and independence results, the family of ultrafilters introduced here should be interesting from combinatorial point of view too.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"69 1","pages":"117-129"},"PeriodicalIF":0.3,"publicationDate":"2023-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/malq.202200070","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50147578","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}

{"title":"The cofinality of the strong measure zero ideal for κ inaccessible","authors":"Johannes Philipp Schürz","doi":"10.1002/malq.202000093","DOIUrl":"https://doi.org/10.1002/malq.202000093","url":null,"abstract":"<p>We investigate the cofinality of the strong measure zero ideal for κ inaccessible and show that it is independent of the size of 2^κ.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"69 1","pages":"31-39"},"PeriodicalIF":0.3,"publicationDate":"2023-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/malq.202000093","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50147576","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}

{"title":"Some definable types that cannot be amalgamated","authors":"Martin Hils,&nbsp;Rosario Mennuni","doi":"10.1002/malq.202200046","DOIUrl":"https://doi.org/10.1002/malq.202200046","url":null,"abstract":"<p>We exhibit a theory where definable types lack the amalgamation property.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"69 1","pages":"46-49"},"PeriodicalIF":0.3,"publicationDate":"2023-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/malq.202200046","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50147577","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}

{"title":"The subset relation and 2-stratified sentences in set theory and class theory","authors":"Zachiri McKenzie","doi":"10.1002/malq.202200029","DOIUrl":"https://doi.org/10.1002/malq.202200029","url":null,"abstract":"&lt;p&gt;Hamkins and Kikuchi (2016, 2017) show that in both set theory and class theory the definable subset ordering of the universe interprets a complete and decidable theory. This paper identifies the minimum subsystem of &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;ZF&lt;/mi&gt;\u0000 &lt;annotation&gt;$mathsf {ZF}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;BAS&lt;/mi&gt;\u0000 &lt;annotation&gt;$mathsf {BAS}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, that ensures that the definable subset ordering of the universe interprets a complete theory, and classifies the structures that can be realised as the subset relation in a model of this set theory. Extending and refining Hamkins and Kikuchi's result for class theory, a complete extension, &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;IABA&lt;/mi&gt;\u0000 &lt;mi&gt;Ideal&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;annotation&gt;$mathsf {IABA}_{mathsf {Ideal}}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, of the theory of infinite atomic boolean algebras and a minimum subsystem, &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;BAC&lt;/mi&gt;\u0000 &lt;mo&gt;+&lt;/mo&gt;\u0000 &lt;/msup&gt;\u0000 &lt;annotation&gt;$mathsf {BAC}^+$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, of &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;NBG&lt;/mi&gt;\u0000 &lt;annotation&gt;$mathsf {NBG}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; are identified with the property that if &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;M&lt;/mi&gt;\u0000 &lt;annotation&gt;$mathcal {M}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is a model of &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;BAC&lt;/mi&gt;\u0000 &lt;mo&gt;+&lt;/mo&gt;\u0000 &lt;/msup&gt;\u0000 &lt;annotation&gt;$mathsf {BAC}^+$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, then &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;⟨&lt;/mo&gt;\u0000 &lt;mi&gt;M&lt;/mi&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;S&lt;/mi&gt;\u0000 &lt;mi&gt;M&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;msup&gt;\u0000 &lt;mo&gt;⊆&lt;/mo&gt;\u0000 &lt;mi&gt;M&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 &lt;mo&gt;⟩&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$langle M, mathcal {S}^mathcal {M}, subseteq ^mathcal {M} rangle$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is a model of &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;IABA&lt;/mi&gt;\u0000 &lt;mi&gt;Ideal&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;annotation&gt;$mathsf {IABA}_{mathsf {Ideal}}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, where &lt;i&gt;M&lt;/i&gt; is the underlying set of &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;M&lt;/mi&gt;\u0000 &lt;annotation&gt;$mathcal {M}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 ","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"69 1","pages":"77-91"},"PeriodicalIF":0.3,"publicationDate":"2023-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50147305","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}

{"title":"On splitting trees","authors":"Giorgio Laguzzi,&nbsp;Heike Mildenberger,&nbsp;Brendan Stuber-Rousselle","doi":"10.1002/malq.202200022","DOIUrl":"https://doi.org/10.1002/malq.202200022","url":null,"abstract":"<p>We investigate two variants of splitting tree forcing, their ideals and regularity properties. We prove connections with other well-known notions, such as Lebesgue measurablility, Baire- and Doughnut-property and the Marczewski field. Moreover, we prove that any <i>absolute</i> amoeba forcing for splitting trees necessarily adds a dominating real, providing more support to Hein's and Spinas' conjecture that <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>add</mo>\u0000 <mo>(</mo>\u0000 <msub>\u0000 <mi>I</mi>\u0000 <mi>SP</mi>\u0000 </msub>\u0000 <mo>)</mo>\u0000 <mo>≤</mo>\u0000 <mi>b</mi>\u0000 </mrow>\u0000 <annotation>$operatorname{add}(mathcal {I}_mathbb {SP}) le mathfrak {b}$</annotation>\u0000 </semantics></math>.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"69 1","pages":"15-30"},"PeriodicalIF":0.3,"publicationDate":"2023-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/malq.202200022","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50155350","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}

{"title":"Incomparable \u0000 \u0000 \u0000 V\u0000 γ\u0000 \u0000 $V_gamma$\u0000 -degrees","authors":"Teng Zhang","doi":"10.1002/malq.202200034","DOIUrl":"https://doi.org/10.1002/malq.202200034","url":null,"abstract":"<p>In [3], Shi proved that there exist incomparable Zermelo degrees at γ if there exists an ω-sequence of measurable cardinals, whose limit is γ. He asked whether there is a size <math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>γ</mi>\u0000 <mi>ω</mi>\u0000 </msup>\u0000 <annotation>$gamma ^omega$</annotation>\u0000 </semantics></math> antichain of Zermelo degrees. We consider this question for the <math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>V</mi>\u0000 <mi>γ</mi>\u0000 </msub>\u0000 <annotation>$V_gamma$</annotation>\u0000 </semantics></math>-degree structure. We use a kind of Prikry-type forcing to show that if there is an ω-sequence of measurable cardinals, then there are <math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>γ</mi>\u0000 <mi>ω</mi>\u0000 </msup>\u0000 <annotation>$gamma ^omega$</annotation>\u0000 </semantics></math>-many pairwise incomparable <math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>V</mi>\u0000 <mi>γ</mi>\u0000 </msub>\u0000 <annotation>$V_gamma$</annotation>\u0000 </semantics></math>-degrees, where γ is the limit of the ω-sequence of measurable cardinals.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"69 1","pages":"58-62"},"PeriodicalIF":0.3,"publicationDate":"2023-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50154667","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}

{"title":"Strongly unfoldable, splitting and bounding","authors":"Ömer Faruk Bağ,&nbsp;Vera Fischer","doi":"10.1002/malq.202200003","DOIUrl":"https://doi.org/10.1002/malq.202200003","url":null,"abstract":"<p>Assuming <math>\u0000 <semantics>\u0000 <mi>GCH</mi>\u0000 <annotation>$mathsf {GCH}$</annotation>\u0000 </semantics></math>, we show that generalized eventually narrow sequences on a strongly inaccessible cardinal κ are preserved under a one step iteration of the Hechler forcing for adding a dominating κ-real. Moreover, we show that if κ is strongly unfoldable, <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mn>2</mn>\u0000 <mi>κ</mi>\u0000 </msup>\u0000 <mo>=</mo>\u0000 <msup>\u0000 <mi>κ</mi>\u0000 <mo>+</mo>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$2^kappa =kappa ^+$</annotation>\u0000 </semantics></math> and λ is a regular cardinal such that <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mi>κ</mi>\u0000 <mo>+</mo>\u0000 </msup>\u0000 <mo>&lt;</mo>\u0000 <mi>λ</mi>\u0000 </mrow>\u0000 <annotation>$kappa ^+ &lt; lambda$</annotation>\u0000 </semantics></math>, then there is a set generic extension in which <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>s</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>κ</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <mo>=</mo>\u0000 <msup>\u0000 <mi>κ</mi>\u0000 <mo>+</mo>\u0000 </msup>\u0000 <mo>&lt;</mo>\u0000 <mi>b</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>κ</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <mo>=</mo>\u0000 <mi>c</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>κ</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <mo>=</mo>\u0000 <mi>λ</mi>\u0000 </mrow>\u0000 <annotation>$mathfrak {s}(kappa ) = kappa ^+ &lt; mathfrak {b}(kappa ) = mathfrak {c}(kappa ) = lambda$</annotation>\u0000 </semantics></math>.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"69 1","pages":"7-14"},"PeriodicalIF":0.3,"publicationDate":"2023-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/malq.202200003","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50154033","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}

{"title":"A note on \u0000 \u0000 fsg\u0000 $text{fsg}$\u0000 groups in p-adically closed fields","authors":"Will Johnson","doi":"10.1002/malq.202200026","DOIUrl":"https://doi.org/10.1002/malq.202200026","url":null,"abstract":"<p>Let <i>G</i> be a definable group in a <i>p</i>-adically closed field <i>M</i>. We show that <i>G</i> has finitely satisfiable generics (<math>\u0000 <semantics>\u0000 <mtext>fsg</mtext>\u0000 <annotation>$text{fsg}$</annotation>\u0000 </semantics></math>) if and only if <i>G</i> is definably compact. The case <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>M</mi>\u0000 <mo>=</mo>\u0000 <msub>\u0000 <mi>Q</mi>\u0000 <mi>p</mi>\u0000 </msub>\u0000 </mrow>\u0000 <annotation>$M = mathbb {Q}_p$</annotation>\u0000 </semantics></math> was previously proved by Onshuus and Pillay.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"69 1","pages":"50-57"},"PeriodicalIF":0.3,"publicationDate":"2023-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50154036","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}

{"title":"Decomposition into special submanifolds","authors":"Masato Fujita","doi":"10.1002/malq.202200057","DOIUrl":"https://doi.org/10.1002/malq.202200057","url":null,"abstract":"<p>We study definably complete locally o-minimal expansions of ordered groups. We propose a notion of special submanifolds with tubular neighborhoods and show that any definable set is decomposed into finitely many special submanifolds with tubular neighborhoods.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"69 1","pages":"104-116"},"PeriodicalIF":0.3,"publicationDate":"2023-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50116854","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}