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Failure of cut-elimination in cyclic-proof systems of logic of bunched implications with inductive propositions 带归纳命题的集束蕴涵逻辑循环证明系统的切消失效
IF 0.4 4区 数学
Archive for Mathematical Logic Pub Date : 2025-11-21 DOI: 10.1007/s00153-025-00993-2
Kenji Saotome, Koji Nakazawa, Daisuke Kimura, Ayumu Kawasaki
{"title":"Failure of cut-elimination in cyclic-proof systems of logic of bunched implications with inductive propositions","authors":"Kenji Saotome,&nbsp;Koji Nakazawa,&nbsp;Daisuke Kimura,&nbsp;Ayumu Kawasaki","doi":"10.1007/s00153-025-00993-2","DOIUrl":"10.1007/s00153-025-00993-2","url":null,"abstract":"<div><p>Cyclic-proof systems are sequent-calculus-style proof systems that allow circular structures that represent induction. Cyclic-proof systems are considered suitable for automated inductive reasoning, and the cut-elimination property is desirable because finding cut formulas often requires heuristics. However, the cut-elimination property does not hold in some cyclic-proof systems, such as the sequent calculus for first-order logic and the entailment system for symbolic-heap separation logic. This paper proves that the cyclic-proof system for the logic of bunched implications does not satisfy the cut-elimination property, even if the system is restricted to the positive fragment with inductively defined propositions. To prove this, we use a new proof technique called proof unrolling. To demonstrate that proof unrolling is a general technique, this paper adapts proof unrolling to another cyclic-proof system for multiplicative additive linear logic with fixed-point operators.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"65 3","pages":"333 - 362"},"PeriodicalIF":0.4,"publicationDate":"2025-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00153-025-00993-2.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147617980","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}
引用次数: 0
On the hierarchy of linearity axioms 关于线性公理的层次
IF 0.4 4区 数学
Archive for Mathematical Logic Pub Date : 2025-11-19 DOI: 10.1007/s00153-025-00999-w
Makoto Fujiwara
{"title":"On the hierarchy of linearity axioms","authors":"Makoto Fujiwara","doi":"10.1007/s00153-025-00999-w","DOIUrl":"10.1007/s00153-025-00999-w","url":null,"abstract":"<div><p>We consider six variations of the linearity axioms along with well-known logical axioms. Then we completely solve both of the hierarchy of the intermediate propositional logics and the hierarchy of semi-classical arithmetic with respect to those logical axioms. In particular, the schema of <span>(Sigma _{n})</span>-substitutions of a variation of the linearity axiom forms a new class in the hierarchy of semi-classical arithmetic. Our investigations reveal that the difference between the acceptable rules for the entailment relation for intermediate propositional logics and those for arithmetic sometimes yields a crucial difference in the hierarchy.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"65 3","pages":"319 - 331"},"PeriodicalIF":0.4,"publicationDate":"2025-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147618015","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}
引用次数: 0
Apartness and the elimination of strong forms of extensionality 分离和消除强形式的延伸性
IF 0.4 4区 数学
Archive for Mathematical Logic Pub Date : 2025-11-16 DOI: 10.1007/s00153-025-00992-3
Benno van den Berg
{"title":"Apartness and the elimination of strong forms of extensionality","authors":"Benno van den Berg","doi":"10.1007/s00153-025-00992-3","DOIUrl":"10.1007/s00153-025-00992-3","url":null,"abstract":"<div><p>We introduce a new version of arithmetic in all finite types which extends the usual versions with primitive notions of extensionality and extensional equality. This new hybrid version allows us to formulate a strong form of extensionality, which we call converse extensionality. Inspired by Brouwer’s notion of apartness, we show that converse extensionality can be eliminated in a way which improves on results from our previous work. We also explain how standard proof-theoretic interpretations, like realizability and functional interpretations, can be extended to such hybrid systems, and how that might be relevant to proof-mining.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"65 3","pages":"297 - 317"},"PeriodicalIF":0.4,"publicationDate":"2025-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147617979","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}
引用次数: 0
On sequential theorems in Reverse Mathematics 论逆向数学中的顺序定理
IF 0.4 4区 数学
Archive for Mathematical Logic Pub Date : 2025-11-15 DOI: 10.1007/s00153-025-00991-4
Dag Normann, Sam Sanders
{"title":"On sequential theorems in Reverse Mathematics","authors":"Dag Normann,&nbsp;Sam Sanders","doi":"10.1007/s00153-025-00991-4","DOIUrl":"10.1007/s00153-025-00991-4","url":null,"abstract":"<div><p>Many theorems of mathematics have the form that for a certain <i>problem</i>, e.g. a differential equation or polynomial (in)equality, there exists a <i>solution</i>. The <i>sequential</i> version then states that for a <i>sequence</i> of problems, there is a <i>sequence</i> of solutions. The original and sequential theorem can often be proved via the same (or similar) proof and often have the same (or similar) logical properties, esp. if everything is formulated in the language of second-order arithmetic. In this paper, we identify basic theorems of third-order arithmetic, e.g. concerning semi-continuous functions, such that the sequential versions have very different logical properties. In particular, depending on the constructive status of the original theorem, very different and independent choice principles are needed. Despite these differences, the associated Reverse Mathematics, working in Kohlenbach’s higher-order framework, is rather elegant and is still based at the core on <i>weak König’s lemma</i>.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"65 2","pages":"275 - 295"},"PeriodicalIF":0.4,"publicationDate":"2025-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147339178","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}
引用次数: 0
On the hausdorff dimension of maximal chains and antichains of turing and hyperarithmetic degrees 图灵和超算术度的极大链和反链的豪斯多夫维数
IF 0.4 4区 数学
Archive for Mathematical Logic Pub Date : 2025-11-09 DOI: 10.1007/s00153-025-00995-0
Sirun Song, Liang Yu
{"title":"On the hausdorff dimension of maximal chains and antichains of turing and hyperarithmetic degrees","authors":"Sirun Song,&nbsp;Liang Yu","doi":"10.1007/s00153-025-00995-0","DOIUrl":"10.1007/s00153-025-00995-0","url":null,"abstract":"<div><p>This paper investigates the Hausdorff dimension properties of chains and antichains in Turing degrees and hyperarithmetic degrees. Our main contributions are threefold: First, for antichains in hyperarithmetic degrees, we prove that every maximal antichain necessarily attains Hausdorff dimension 1. Second, regarding chains in Turing degrees, we establish the existence of a maximal chain with Hausdorff dimension 0. Furthermore, under the assumption that <span>(omega _1=(omega _1)^L)</span>, we demonstrate the existence of such maximal chains with <span>(Pi ^1_1)</span> complexity. Third, we extend our investigation to maximal antichains of Turing degrees by analyzing both the packing dimension and effective Hausdorff dimension.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"65 2","pages":"213 - 238"},"PeriodicalIF":0.4,"publicationDate":"2025-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147338356","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}
引用次数: 0
Results in model theory for Riesz-valued structures riesz值结构的模型理论结果
IF 0.4 4区 数学
Archive for Mathematical Logic Pub Date : 2025-11-08 DOI: 10.1007/s00153-025-00990-5
Stefano Baratella
{"title":"Results in model theory for Riesz-valued structures","authors":"Stefano Baratella","doi":"10.1007/s00153-025-00990-5","DOIUrl":"10.1007/s00153-025-00990-5","url":null,"abstract":"<div><p>We study model-theoretic properties of a logic whose formulas take values in suitable Riesz spaces. In addition to having a set of truth values which is not linearly ordered, there is no absolute truth or falsehood value. We take inspiration from the version of continuous logic developed in [8] by Ben-Yaacov <i>et al.</i> and from the more general approach recently proposed in [11] by Keisler. We extend to our framework a number of results obtained by the above mentioned authors. Under suitable assumptions on the underlying Riesz space, we provide an ultraproduct construction and prove a Łoś theorem, from which compactness follows. In the framework of Riesz spaces we also address definability issues and we extend metric notions, obtaining analogues of Keisler’s pre-metric structures and pre-metric expansions of theories defined in Keisler. In particular, we prove that every theory in a countable language has a pre-metric expansion with a so-called pseudometric approximate distance.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"65 2","pages":"239 - 273"},"PeriodicalIF":0.4,"publicationDate":"2025-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00153-025-00990-5.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147337789","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}
引用次数: 0
Around the Eggleston Theorem 根据埃格尔斯顿定理
IF 0.4 4区 数学
Archive for Mathematical Logic Pub Date : 2025-11-07 DOI: 10.1007/s00153-025-00998-x
Marcin Michalski, Robert Rałowski, Szymon Żeberski
{"title":"Around the Eggleston Theorem","authors":"Marcin Michalski,&nbsp;Robert Rałowski,&nbsp;Szymon Żeberski","doi":"10.1007/s00153-025-00998-x","DOIUrl":"10.1007/s00153-025-00998-x","url":null,"abstract":"<div><p>This work is motivated by the two classical theorems on inscribing rectangles and squares into large subsets of the plane: Eggleston Theorem and Mycielski Theorem. Using Shoenfield Absoluteness Theorem, we prove that for every Borel subset of the plane with uncountably many vertical sections that are positive (with respect to measure or category) contains a rectangle <span>(Ptimes B)</span> where <i>P</i> is perfect and <i>B</i> is Borel and positive. We also present a variant of Eggleston Theorem regarding the <span>(sigma -)</span>ideal <span>(mathcal {E})</span> generated by closed sets of measure zero. Furthermore, we show that every comeager (resp. conull) subset of the plane contains a rectangle <span>([T]times H)</span>, where <i>T</i> is a splitting tree containing a Silver tree and <i>H</i> is comeager (resp. conull). Additionally, we establish a joint generalization of Eggleston Theorem and Mycielski Theorem stating that every comeager (resp. conull) subset of the plane contains a rectangle <span>([T]times H)</span> modulo diagonal, where <i>T</i> is a uniformly perfect tree, <i>H</i> is comeager (resp. conull) and <span>([T]subseteq H)</span>.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"65 2","pages":"193 - 212"},"PeriodicalIF":0.4,"publicationDate":"2025-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00153-025-00998-x.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147337770","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}
引用次数: 0
Nonvanishing derived limits without scales 无尺度的不消失的派生极限。
IF 0.4 4区 数学
Archive for Mathematical Logic Pub Date : 2025-11-04 DOI: 10.1007/s00153-025-00996-z
Matteo Casarosa
{"title":"Nonvanishing derived limits without scales","authors":"Matteo Casarosa","doi":"10.1007/s00153-025-00996-z","DOIUrl":"10.1007/s00153-025-00996-z","url":null,"abstract":"<div><p>The derived functors <span>(lim ^n)</span> of the inverse limit are widely studied for their topological applications, among which are some repercussions on the additivity of strong homology. Set theory has proven useful in dealing with these functors, for instance in the case of the inverse system <span>(textbf{A})</span> of abelian groups indexed over <span>({}^omega omega )</span>. So far, consistency results for nonvanishing derived limits of <span>(textbf{A})</span> have always assumed the existence of a scale (i.e. a linear cofinal subset of <span>(({}^omega omega , le ^*))</span>, or equivalently that <span>(mathfrak {b} = mathfrak {d} )</span>). Here we eliminate that assumption and prove that nonvanishing derived limits, and hence the non-additivity of strong homology, are consistent with any value of <span>(aleph _1 le mathfrak {b} le mathfrak {d} &lt; aleph _omega )</span>, thus giving a partial answer to a question of Bannister.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"65 2","pages":"177 - 191"},"PeriodicalIF":0.4,"publicationDate":"2025-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12881080/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146140904","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}
引用次数: 0
Homogeneity, P-like properties of ideals and orders between ideals 同质性，理想的类p性质以及理想间的顺序
IF 0.4 4区 数学
Archive for Mathematical Logic Pub Date : 2025-11-02 DOI: 10.1007/s00153-025-00994-1
Krzysztof Kowitz, Jacek Tryba
{"title":"Homogeneity, P-like properties of ideals and orders between ideals","authors":"Krzysztof Kowitz,&nbsp;Jacek Tryba","doi":"10.1007/s00153-025-00994-1","DOIUrl":"10.1007/s00153-025-00994-1","url":null,"abstract":"<div><p>We give characterizations for the thin ideal and almost thin ideal, which were studied by Jörg Brendle and Jana Flašková in their paper Brendle, J. <b>236</b>(3), 201–245 (2017). We check the various properties of selected ideals. We also verify whether these ideals are comparable in the Rudin-Blass order and Rudin-Keisler order. Moreover, we present a negative answer to [Filipów, R. 1–48 (2024) Question 8.8(1)].</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"65 2","pages":"151 - 175"},"PeriodicalIF":0.4,"publicationDate":"2025-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00153-025-00994-1.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147335788","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}
引用次数: 0
Constructible structures and constructible open cores 可建造的结构和可建造的开芯
IF 0.4 4区 数学
Archive for Mathematical Logic Pub Date : 2025-10-30 DOI: 10.1007/s00153-025-00989-y
Masato Fujita
{"title":"Constructible structures and constructible open cores","authors":"Masato Fujita","doi":"10.1007/s00153-025-00989-y","DOIUrl":"10.1007/s00153-025-00989-y","url":null,"abstract":"<div><p>We give a sufficient condition for definably complete expansions <span>({mathcal {R}})</span> of ordered groups to have constructible open cores. We also prove that the dimension function satisfies the axioms proposed by van den Dries if <span>({mathcal {R}})</span> is Baire and every set definable in <span>({mathcal {R}})</span> is constructible. Using the first result, we give necessary and sufficient conditions for definably complete ordered groups (or definably complete ordered fields) to have locally o-minimal (or d-minimal) open cores. As a corollary, we obtain that, if the theory of <span>({mathcal {R}})</span> is strong, <span>({mathcal {R}})</span> has a locally o-minimal open core.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"65 2","pages":"133 - 149"},"PeriodicalIF":0.4,"publicationDate":"2025-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147342918","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}
引用次数: 0
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