Mathematical Logic Quarterly最新文献

{"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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}

{"title":"The power set and the set of permutations with finitely many non-fixed points of a set","authors":"Guozhen Shen","doi":"10.1002/malq.202100070","DOIUrl":"https://doi.org/10.1002/malq.202100070","url":null,"abstract":"<p>For a cardinal <math>\u0000 <semantics>\u0000 <mi>a</mi>\u0000 <annotation>$mathfrak {a}$</annotation>\u0000 </semantics></math>, we write <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mrow>\u0000 <msub>\u0000 <mi>S</mi>\u0000 <mtext>fin</mtext>\u0000 </msub>\u0000 </mrow>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>a</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$operatorname{mathcal {S}_{text{fin}}}(mathfrak {a})$</annotation>\u0000 </semantics></math> for the cardinality of the set of permutations with finitely many non-fixed points of a set which is of cardinality <math>\u0000 <semantics>\u0000 <mi>a</mi>\u0000 <annotation>$mathfrak {a}$</annotation>\u0000 </semantics></math>. We investigate the relationships between <math>\u0000 <semantics>\u0000 <msup>\u0000 <mn>2</mn>\u0000 <mi>a</mi>\u0000 </msup>\u0000 <annotation>$2^mathfrak {a}$</annotation>\u0000 </semantics></math> and <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mrow>\u0000 <msub>\u0000 <mi>S</mi>\u0000 <mtext>fin</mtext>\u0000 </msub>\u0000 </mrow>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>a</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$operatorname{mathcal {S}_{text{fin}}}(mathfrak {a})$</annotation>\u0000 </semantics></math> for an arbitrary infinite cardinal <math>\u0000 <semantics>\u0000 <mi>a</mi>\u0000 <annotation>$mathfrak {a}$</annotation>\u0000 </semantics></math> in <math>\u0000 <semantics>\u0000 <mi>ZF</mi>\u0000 <annotation>$mathsf {ZF}$</annotation>\u0000 </semantics></math> (without the axiom of choice). It is proved in <math>\u0000 <semantics>\u0000 <mi>ZF</mi>\u0000 <annotation>$mathsf {ZF}$</annotation>\u0000 </semantics></math> that <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mn>2</mn>\u0000 <mi>a</mi>\u0000 </msup>\u0000 <mo>≠</mo>\u0000 <mrow>\u0000 <msub>\u0000 <mi>S</mi>\u0000 <mtext>fin</mtext>\u0000 </msub>\u0000 </mrow>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>a</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$2^mathfrak {a}ne operatorname{mathcal {S}_{text{fin}}}(mathfrak {a})$</annotation>\u0000 </semantics></math> for all inf","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2023-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50127714","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":"Contents: (Math. Log. Quart. 4/2022)","authors":"","doi":"10.1002/malq.202240000","DOIUrl":"10.1002/malq.202240000","url":null,"abstract":"","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2022-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/malq.202240000","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80624922","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":"Piece selection and cardinal arithmetic","authors":"Pierre Matet","doi":"10.1002/malq.202100033","DOIUrl":"10.1002/malq.202100033","url":null,"abstract":"<p>We study the effects of piece selection principles on cardinal arithmetic (Shelah style). As an application, we discuss questions of Abe and Usuba. In particular, we show that if <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>λ</mi>\u0000 <mo>≥</mo>\u0000 <msup>\u0000 <mn>2</mn>\u0000 <mi>κ</mi>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$lambda ge 2^kappa$</annotation>\u0000 </semantics></math>, then (a) <math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>I</mi>\u0000 <mrow>\u0000 <mi>κ</mi>\u0000 <mo>,</mo>\u0000 <mi>λ</mi>\u0000 </mrow>\u0000 </msub>\u0000 <annotation>$I_{kappa , lambda }$</annotation>\u0000 </semantics></math> is not (λ, 2)-distributive, and (b) <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msubsup>\u0000 <mi>I</mi>\u0000 <mrow>\u0000 <mi>κ</mi>\u0000 <mo>,</mo>\u0000 <mi>λ</mi>\u0000 </mrow>\u0000 <mo>+</mo>\u0000 </msubsup>\u0000 <mo>→</mo>\u0000 <msubsup>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <msubsup>\u0000 <mi>I</mi>\u0000 <mrow>\u0000 <mi>κ</mi>\u0000 <mo>,</mo>\u0000 <mi>λ</mi>\u0000 </mrow>\u0000 <mo>+</mo>\u0000 </msubsup>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <mi>ω</mi>\u0000 <mn>2</mn>\u0000 </msubsup>\u0000 </mrow>\u0000 <annotation>$I_{kappa , lambda }^+ rightarrow (I_{kappa , lambda }^+)^2_omega$</annotation>\u0000 </semantics></math> does not hold.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2022-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74802387","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":"Intuitionistic propositional probability logic","authors":"Anelina Ilić-Stepić,&nbsp;Mateja Knežević,&nbsp;Zoran Ognjanović","doi":"10.1002/malq.202100052","DOIUrl":"10.1002/malq.202100052","url":null,"abstract":"<p>We give a sound and complete axiomatization of a probabilistic extension of intuitionistic logic. Reasoning with probability operators is also intuitionistic (in contradistinction to other works on this topic), i.e., measure functions used for modeling probability operators are partial functions. Finally, we present a decision procedure for our logic, which is a combination of linear programming and an intuitionistic tableaux method.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2022-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83342708","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":"Automorphism invariant measures and weakly generic automorphisms","authors":"Gábor Sági","doi":"10.1002/malq.202100044","DOIUrl":"10.1002/malq.202100044","url":null,"abstract":"<p>Let <math>\u0000 <semantics>\u0000 <mi>A</mi>\u0000 <annotation>$mathcal {A}$</annotation>\u0000 </semantics></math> be a countable ℵ₀-homogeneous structure. The primary motivation of this work is to study different amenability properties of (subgroups of) the automorphism group <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>Aut</mo>\u0000 <mo>(</mo>\u0000 <mi>A</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$operatorname{Aut}(mathcal {A})$</annotation>\u0000 </semantics></math> of <math>\u0000 <semantics>\u0000 <mi>A</mi>\u0000 <annotation>$mathcal {A}$</annotation>\u0000 </semantics></math>; the secondary motivation is to study the existence of weakly generic automorphisms of <math>\u0000 <semantics>\u0000 <mi>A</mi>\u0000 <annotation>$mathcal {A}$</annotation>\u0000 </semantics></math>. Among others, we present sufficient conditions implying the existence of automorphism invariant probability measures on certain subsets of <i>A</i> and of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>Aut</mo>\u0000 <mo>(</mo>\u0000 <mi>A</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$operatorname{Aut}(mathcal {A})$</annotation>\u0000 </semantics></math>; we also present sufficient conditions implying that the theory of <math>\u0000 <semantics>\u0000 <mi>A</mi>\u0000 <annotation>$mathcal {A}$</annotation>\u0000 </semantics></math> is amenable. More concretely, we show that if the set of locally finite automorphisms of <math>\u0000 <semantics>\u0000 <mi>A</mi>\u0000 <annotation>$mathcal {A}$</annotation>\u0000 </semantics></math> is dense (in particular, if <math>\u0000 <semantics>\u0000 <mi>A</mi>\u0000 <annotation>$mathcal {A}$</annotation>\u0000 </semantics></math> has weakly generic tuples of automorphisms of arbitrary finite length), then there exists a finitely additive probability measure μ on the subsets of <math>\u0000 <semantics>\u0000 <mi>A</mi>\u0000 <annotation>$mathcal {A}$</annotation>\u0000 </semantics></math> definable with parameters such that μ is invariant under <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>Aut</mo>\u0000 <mo>(</mo>\u0000 <mi>A</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$operatorname{Aut}(mathcal {A})$</annotation>\u0000 </semantics></math>. Moreover, if <math>\u0000 <semantics>\u0000 <mi>A</mi>\u0000 <annotation>$mathcal {A}","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2022-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/malq.202100044","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85168891","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}