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{"title":"UNDEFINABILITY OF MULTIPLICATION IN PRESBURGER ARITHMETIC WITH SETS OF POWERS","authors":"CHRIS SCHULZ","doi":"10.1017/jsl.2023.71","DOIUrl":"https://doi.org/10.1017/jsl.2023.71","url":null,"abstract":"An abstract is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.","PeriodicalId":17088,"journal":{"name":"Journal of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136295795","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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{"title":"FINITE UNDECIDABILITY IN NIP FIELDS","authors":"BRIAN TYRRELL","doi":"10.1017/jsl.2023.73","DOIUrl":"https://doi.org/10.1017/jsl.2023.73","url":null,"abstract":"An abstract is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.","PeriodicalId":17088,"journal":{"name":"Journal of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135591953","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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{"title":"Weak Indestructibility and Reflection","authors":"James Holland","doi":"10.1017/jsl.2023.72","DOIUrl":"https://doi.org/10.1017/jsl.2023.72","url":null,"abstract":"An abstract is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.","PeriodicalId":17088,"journal":{"name":"Journal of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135592760","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract The notion of countable well order admits an alternative definition in terms of embeddings between initial segments. We use the framework of reverse mathematics to investigate the logical strength of this definition and its connection with Fraïssé’s conjecture, which has been proved by Laver. We also fill a small gap in Shore’s proof that Fraïssé’s conjecture implies arithmetic transfinite recursion over $mathbf {RCA}_0$ , by giving a new proof of $Sigma ^0_2$ -induction.
{"title":"WEAK WELL ORDERS AND FRAÏSSÉ’S CONJECTURE","authors":"ANTON FREUND, DAVIDE MANCA","doi":"10.1017/jsl.2023.70","DOIUrl":"https://doi.org/10.1017/jsl.2023.70","url":null,"abstract":"Abstract The notion of countable well order admits an alternative definition in terms of embeddings between initial segments. We use the framework of reverse mathematics to investigate the logical strength of this definition and its connection with Fraïssé’s conjecture, which has been proved by Laver. We also fill a small gap in Shore’s proof that Fraïssé’s conjecture implies arithmetic transfinite recursion over $mathbf {RCA}_0$ , by giving a new proof of $Sigma ^0_2$ -induction.","PeriodicalId":17088,"journal":{"name":"Journal of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135537916","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Given a sound first-order p-time theory T capable of formalizing syntax of first-order logic we define a p-time function $g_T$ that stretches all inputs by one bit and we use its properties to show that T must be incomplete. We leave it as an open problem whether for some T the range of $g_T$ intersects all infinite ${mbox {NP}}$ sets (i.e., whether it is a proof complexity generator hard for all proof systems). A propositional version of the construction shows that at least one of the following three statements is true: 1. There is no p-optimal propositional proof system (this is equivalent to the non-existence of a time-optimal propositional proof search algorithm). 2. $E not subseteq P/poly$ . 3. There exists function h that stretches all inputs by one bit, is computable in sub-exponential time, and its range $Rng(h)$ intersects all infinite ${text {NP}}$ sets.
摘要给定一个可靠的一阶p时间理论T，能够形式化一阶逻辑的语法，我们定义了一个p时间函数$g_T$，它将所有输入延伸1位，并利用它的性质证明了T一定是不完全的。对于某些T， $g_T$的范围是否与所有无限的${mbox {NP}}$集合相交(即，它是否是一个对所有证明系统都很难的证明复杂性生成器)，我们将其作为一个开放问题。该结构的命题版本表明，以下三个陈述中至少有一个是正确的:不存在p最优命题证明系统(这相当于不存在时间最优命题证明搜索算法)。2. $E not subseteq P/poly$。3.存在一个函数h，它将所有输入延展1位，在次指数时间内可计算，其值域$Rng(h)$与所有无限的${text {NP}}$集合相交。
{"title":"A proof complexity conjecture and the Incompleteness theorem","authors":"Jan Krajíček","doi":"10.1017/jsl.2023.69","DOIUrl":"https://doi.org/10.1017/jsl.2023.69","url":null,"abstract":"Abstract Given a sound first-order p-time theory T capable of formalizing syntax of first-order logic we define a p-time function $g_T$ that stretches all inputs by one bit and we use its properties to show that T must be incomplete. We leave it as an open problem whether for some T the range of $g_T$ intersects all infinite ${mbox {NP}}$ sets (i.e., whether it is a proof complexity generator hard for all proof systems). A propositional version of the construction shows that at least one of the following three statements is true: 1. There is no p-optimal propositional proof system (this is equivalent to the non-existence of a time-optimal propositional proof search algorithm). 2. $E not subseteq P/poly$ . 3. There exists function h that stretches all inputs by one bit, is computable in sub-exponential time, and its range $Rng(h)$ intersects all infinite ${text {NP}}$ sets.","PeriodicalId":17088,"journal":{"name":"Journal of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135015758","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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{"title":"CLASSIFICATION OF <i>ω</i>-CATEGORICAL MONADICALLY STABLE STRUCTURES","authors":"BERTALAN BODOR","doi":"10.1017/jsl.2023.66","DOIUrl":"https://doi.org/10.1017/jsl.2023.66","url":null,"abstract":"An abstract is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.","PeriodicalId":17088,"journal":{"name":"Journal of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135060292","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We introduce a homotopy-theoretic interpretation of intuitionistic first-order logic based on ideas from Homotopy Type Theory. We provide a categorical formulation of this interpretation using the framework of Grothendieck fibrations. We then use this formulation to prove the central property of this interpretation, namely homotopy invariance. To do this, we use the result from arXiv:1905.10690 that any Grothendieck fibration of the kind being considered can automatically be upgraded to a 2-dimensional fibration, after which the invariance property is reduced to an abstract theorem concerning pseudonatural transformations of morphisms into 2-dimensional fibrations.
{"title":"First-order homotopical logic","authors":"Joseph Helfer","doi":"10.1017/jsl.2023.68","DOIUrl":"https://doi.org/10.1017/jsl.2023.68","url":null,"abstract":"We introduce a homotopy-theoretic interpretation of intuitionistic first-order logic based on ideas from Homotopy Type Theory. We provide a categorical formulation of this interpretation using the framework of Grothendieck fibrations. We then use this formulation to prove the central property of this interpretation, namely homotopy invariance. To do this, we use the result from arXiv:1905.10690 that any Grothendieck fibration of the kind being considered can automatically be upgraded to a 2-dimensional fibration, after which the invariance property is reduced to an abstract theorem concerning pseudonatural transformations of morphisms into 2-dimensional fibrations.","PeriodicalId":17088,"journal":{"name":"Journal of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135110703","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
HEER TERN KOH, ALEXANDER G. MELNIKOV, KENG MENG NG
{"title":"COMPUTABLE TOPOLOGICAL GROUPS","authors":"HEER TERN KOH, ALEXANDER G. MELNIKOV, KENG MENG NG","doi":"10.1017/jsl.2023.67","DOIUrl":"https://doi.org/10.1017/jsl.2023.67","url":null,"abstract":"","PeriodicalId":17088,"journal":{"name":"Journal of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135154380","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract It is consistent relative to an inaccessible cardinal that ZF+DC holds, and the hypergraph of isosceles triangles on $mathbb {R}^2$ has countable chromatic number while the hypergraph of isosceles triangles on $mathbb {R}^3$ has uncountable chromatic number.
{"title":"Coloring isosceles triangles in choiceless set theory","authors":"Yuxin Zhou","doi":"10.1017/jsl.2023.63","DOIUrl":"https://doi.org/10.1017/jsl.2023.63","url":null,"abstract":"Abstract It is consistent relative to an inaccessible cardinal that ZF+DC holds, and the hypergraph of isosceles triangles on $mathbb {R}^2$ has countable chromatic number while the hypergraph of isosceles triangles on $mathbb {R}^3$ has uncountable chromatic number.","PeriodicalId":17088,"journal":{"name":"Journal of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135980726","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We discuss the role of weakly normal formulas in the theory of thorn forking, as part of a commentary on the paper [5]. We also give a counterexample to Corollary 4.2 from that paper, and in the process discuss “theories with selectors.”
{"title":"Thorn forking, weak normality, and theories with selectors","authors":"Daniel Max Hoffmann, Anand Pillay","doi":"10.1017/jsl.2023.64","DOIUrl":"https://doi.org/10.1017/jsl.2023.64","url":null,"abstract":"Abstract We discuss the role of weakly normal formulas in the theory of thorn forking, as part of a commentary on the paper [5]. We also give a counterexample to Corollary 4.2 from that paper, and in the process discuss “theories with selectors.”","PeriodicalId":17088,"journal":{"name":"Journal of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135938631","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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