Mathematical Logic Quarterly最新文献

{"title":"Pregeometry over locally o-minimal structures and dimension","authors":"Masato Fujita","doi":"10.1002/malq.202200069","DOIUrl":"10.1002/malq.202200069","url":null,"abstract":"<p>We define a discrete closure operator for definably complete locally o-minimal structures <math>\u0000 <semantics>\u0000 <mi>M</mi>\u0000 <annotation>$mathcal {M}$</annotation>\u0000 </semantics></math>. The pair of the underlying set of <math>\u0000 <semantics>\u0000 <mi>M</mi>\u0000 <annotation>$mathcal {M}$</annotation>\u0000 </semantics></math> and the discrete closure operator forms a pregeometry. We define the rank of a definable set over a set of parameters using this fact and call it <math>\u0000 <semantics>\u0000 <mo>discl</mo>\u0000 <annotation>$operatorname{discl}$</annotation>\u0000 </semantics></math>-dimension. A definable set <i>X</i> is of dimension equal to the <math>\u0000 <semantics>\u0000 <mo>discl</mo>\u0000 <annotation>$operatorname{discl}$</annotation>\u0000 </semantics></math>-dimension of <i>X</i>. The structure <math>\u0000 <semantics>\u0000 <mi>M</mi>\u0000 <annotation>$mathcal {M}$</annotation>\u0000 </semantics></math> is simultaneously a first-order topological structure. The dimension rank of a set definable in the first-order topological structure <math>\u0000 <semantics>\u0000 <mi>M</mi>\u0000 <annotation>$mathcal {M}$</annotation>\u0000 </semantics></math> also coincides with its dimension.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2023-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76113023","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. 3/2023)","authors":"","doi":"10.1002/malq.202330001","DOIUrl":"https://doi.org/10.1002/malq.202330001","url":null,"abstract":"","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2023-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/malq.202330001","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50146478","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":"Contents: (Math. Log. Quart. 2/2023)","authors":"","doi":"10.1002/malq.202320001","DOIUrl":"https://doi.org/10.1002/malq.202320001","url":null,"abstract":"","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2023-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/malq.202320001","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50138137","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":"Bisimulations and bisimulation games between Verbrugge models","authors":"Sebastijan Horvat,&nbsp;Tin Perkov,&nbsp;Mladen Vuković","doi":"10.1002/malq.202200042","DOIUrl":"https://doi.org/10.1002/malq.202200042","url":null,"abstract":"<p>Interpretability logic is a modal formalization of relative interpretability between first-order arithmetical theories. Verbrugge semantics is a generalization of Veltman semantics, the basic semantics for interpretability logic. Bisimulation is the basic equivalence between models for modal logic. We study various notions of bisimulation between Verbrugge models and develop a new one, which we call w-bisimulation. We show that the new notion, while keeping the basic property that bisimilarity implies modal equivalence, is weak enough to allow the converse to hold in the finitary case. To do this, we develop and use an appropriate notion of bisimulation games between Verbrugge models.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","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":"50126665","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 permutations with n non-fixed points and the subsets with n elements of a set","authors":"Supakun Panasawatwong,&nbsp;Pimpen Vejjajiva","doi":"10.1002/malq.202300005","DOIUrl":"https://doi.org/10.1002/malq.202300005","url":null,"abstract":"<p>We write <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>S</mi>\u0000 <mi>n</mi>\u0000 </msub>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>a</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$mathcal {S}_n(mathfrak {a})$</annotation>\u0000 </semantics></math> and <math>\u0000 <semantics>\u0000 <msup>\u0000 <mrow>\u0000 <mo>[</mo>\u0000 <mi>a</mi>\u0000 <mo>]</mo>\u0000 </mrow>\u0000 <mi>n</mi>\u0000 </msup>\u0000 <annotation>$[mathfrak {a}]^n$</annotation>\u0000 </semantics></math> for the cardinalities of the set of permutations with <i>n</i> non-fixed points and the set of subsets with <i>n</i> elements, respectively, of a set which is of cardinality <math>\u0000 <semantics>\u0000 <mi>a</mi>\u0000 <annotation>$mathfrak {a}$</annotation>\u0000 </semantics></math>, where <i>n</i> is a natural number greater than 1. With the Axiom of Choice, <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>S</mi>\u0000 <mi>n</mi>\u0000 </msub>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>a</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$mathcal {S}_n(mathfrak {a})$</annotation>\u0000 </semantics></math> and <math>\u0000 <semantics>\u0000 <msup>\u0000 <mrow>\u0000 <mo>[</mo>\u0000 <mi>a</mi>\u0000 <mo>]</mo>\u0000 </mrow>\u0000 <mi>n</mi>\u0000 </msup>\u0000 <annotation>$[mathfrak {a}]^n$</annotation>\u0000 </semantics></math> are equal for all infinite cardinals <math>\u0000 <semantics>\u0000 <mi>a</mi>\u0000 <annotation>$mathfrak {a}$</annotation>\u0000 </semantics></math>. We show, in <span>ZF</span>, that if <math>\u0000 <semantics>\u0000 <msub>\u0000 <mtext>AC</mtext>\u0000 <mrow>\u0000 <mo>≤</mo>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 </msub>\u0000 <annotation>$mbox{textsf {AC}}_{le n}$</annotation>\u0000 </semantics></math> is assumed, then <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mrow>\u0000 <mo>[</mo>\u0000 <mi>a</mi>\u0000 <mo>]</mo>\u0000 </mrow>\u0000 <mi>n</mi>\u0000 </msup>\u0000 <mo>≤</mo>\u0000 <msub>\u0000 <mi>S</mi>\u0000 <mi>n</mi>\u0000 </msub>\u0000 ","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","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":"50126677","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 a cardinal inequality in \u0000 \u0000 ZF\u0000 $mathsf {ZF}$","authors":"Guozhen Shen","doi":"10.1002/malq.202300014","DOIUrl":"10.1002/malq.202300014","url":null,"abstract":"<p>It is proved in <math>\u0000 <semantics>\u0000 <mi>ZF</mi>\u0000 <annotation>$mathsf {ZF}$</annotation>\u0000 </semantics></math> (without the axiom of choice) that <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mi>a</mi>\u0000 <mi>n</mi>\u0000 </msup>\u0000 <mo>⩽</mo>\u0000 <msub>\u0000 <mi>S</mi>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 <mo>+</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 </msub>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>a</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$mathfrak {a}^nleqslant mathcal {S}_{n+1}(mathfrak {a})$</annotation>\u0000 </semantics></math> for all infinite cardinals <math>\u0000 <semantics>\u0000 <mi>a</mi>\u0000 <annotation>$mathfrak {a}$</annotation>\u0000 </semantics></math> and all natural numbers <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 <mo>≠</mo>\u0000 <mn>0</mn>\u0000 </mrow>\u0000 <annotation>$nne 0$</annotation>\u0000 </semantics></math>, where <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>S</mi>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 <mo>+</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 </msub>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>a</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$mathcal {S}_{n+1}(mathfrak {a})$</annotation>\u0000 </semantics></math> is the cardinality of the set of permutations with exactly <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 <mo>+</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 <annotation>$n+1$</annotation>\u0000 </semantics></math> 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>.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","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":"87237651","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":"Borel complexity and Ramsey largeness of sets of oracles separating complexity classes","authors":"Alex Creiner,&nbsp;Stephen Jackson","doi":"10.1002/malq.202200068","DOIUrl":"https://doi.org/10.1002/malq.202200068","url":null,"abstract":"<p>We prove two sets of results concerning computational complexity classes. First, we propose a new variation of the random oracle hypothesis, originally posed by Bennett and Gill after they showed that relative to a randomly chosen oracle, <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>P</mi>\u0000 <mo>≠</mo>\u0000 <mi>NP</mi>\u0000 </mrow>\u0000 <annotation>$mathbf {P}ne mathbf {NP}$</annotation>\u0000 </semantics></math> with probability 1. Their original hypothesis was quickly disproven in several ways, most famously in 1992 with the result that <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>IP</mi>\u0000 <mo>=</mo>\u0000 <mi>PSPACE</mi>\u0000 </mrow>\u0000 <annotation>$mathbf {IP} = mathbf {PSPACE}$</annotation>\u0000 </semantics></math>, in spite of the classes being shown unequal with probability 1. Here we propose a variation of what it means to be “large” using the Ellentuck topology. In this new context, we demonstrate that the set of oracles separating <math>\u0000 <semantics>\u0000 <mi>NP</mi>\u0000 <annotation>$mathbf {NP}$</annotation>\u0000 </semantics></math> and <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>co</mi>\u0000 <mi>-</mi>\u0000 <mi>NP</mi>\u0000 </mrow>\u0000 <annotation>$mathbf {co}text{-}mathbf {NP}$</annotation>\u0000 </semantics></math> is not small, and obtain similar results for the separation of <math>\u0000 <semantics>\u0000 <mi>PSPACE</mi>\u0000 <annotation>$mathbf {PSPACE}$</annotation>\u0000 </semantics></math> from <math>\u0000 <semantics>\u0000 <mi>PH</mi>\u0000 <annotation>$mathbf {PH}$</annotation>\u0000 </semantics></math> along with the separation of <math>\u0000 <semantics>\u0000 <mi>NP</mi>\u0000 <annotation>$mathbf {NP}$</annotation>\u0000 </semantics></math> from <math>\u0000 <semantics>\u0000 <mi>BQP</mi>\u0000 <annotation>$mathbf {BQP}$</annotation>\u0000 </semantics></math>. We also show that the set of oracles <i>equating</i> <math>\u0000 <semantics>\u0000 <mi>IP</mi>\u0000 <annotation>$mathbf {IP}$</annotation>\u0000 </semantics></math> with <math>\u0000 <semantics>\u0000 <mi>PSPACE</mi>\u0000 <annotation>$mathbf {PSPACE}$</annotation>\u0000 </semantics></math> is large in this new sense. We demonstrate that this version of the hypothesis provides a sufficient condition for unrelativized relationships, at least in the cases considered here. Second, we examine the descriptive complexity of the classes of oracles providing the separations for these various classes, and determine their exact placement in the Borel hierarchy.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2023-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50128497","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 self-distributive weak Heyting algebras","authors":"Mohsen Nourany,&nbsp;Shokoofeh Ghorbani,&nbsp;Arsham Borumand Saeid","doi":"10.1002/malq.202200073","DOIUrl":"https://doi.org/10.1002/malq.202200073","url":null,"abstract":"<p>We use the left self-distributive axiom to introduce and study a special class of weak Heyting algebras, called self-distributive weak Heyting algebras (SDWH-algebras). We present some useful properties of SDWH-algebras and obtain some equivalent conditions of them. A characteristic of SDWH-algebras of orders 3 and 4 is given. Finally, we study the relation between the variety of SDWH-algebras and some of the known subvarieties of weak Heyting algebras such as the variety of Heyting algebras, the variety of basic algebras, the variety of subresiduated lattices, the variety of reflexive WH-algebras (RWH-algebras), and the variety of transitive WH-algebras (TWH-algebras).</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2023-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50117486","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":"Models of \u0000 \u0000 \u0000 VTC\u0000 0\u0000 \u0000 $mathsf {VTC^0}$\u0000 as exponential integer parts","authors":"Emil Jeřábek","doi":"10.1002/malq.202300001","DOIUrl":"https://doi.org/10.1002/malq.202300001","url":null,"abstract":"<p>We prove that (additive) ordered group reducts of nonstandard models of the bounded arithmetical theory <math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>VTC</mi>\u0000 <mn>0</mn>\u0000 </msup>\u0000 <annotation>$mathsf {VTC^0}$</annotation>\u0000 </semantics></math> are recursively saturated in a rich language with predicates expressing the integers, rationals, and logarithmically bounded numbers. Combined with our previous results on the construction of the real exponential function on completions of models of <math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>VTC</mi>\u0000 <mn>0</mn>\u0000 </msup>\u0000 <annotation>$mathsf {VTC^0}$</annotation>\u0000 </semantics></math>, we show that every countable model of <math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>VTC</mi>\u0000 <mn>0</mn>\u0000 </msup>\u0000 <annotation>$mathsf {VTC^0}$</annotation>\u0000 </semantics></math> is an exponential integer part of a real-closed exponential field.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2023-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/malq.202300001","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50117487","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":"Forcing revisited","authors":"Toby Meadows","doi":"10.1002/malq.202000040","DOIUrl":"https://doi.org/10.1002/malq.202000040","url":null,"abstract":"<p>The purpose of this paper is to propose and explore a general framework within which a wide variety of model construction techniques from contemporary set theory can be subsumed. Taking our inspiration from presheaf constructions in category theory and Boolean ultrapowers, we will show that generic extensions, ultrapowers, extenders and generic ultrapowers can be construed as examples of a single model construction technique. In particular, we will show that Łoś's theorem can be construed as a specific case of Cohen's truth lemma, and we isolate the weakest conditions a filter must satisfy in order for the truth lemma to work.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50115389","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}