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Effective weak and vague convergence of measures on the real line 实线上度量的有效弱收敛和模糊收敛
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2023-09-27 DOI: 10.1007/s00153-023-00886-2
Diego A. Rojas
{"title":"Effective weak and vague convergence of measures on the real line","authors":"Diego A. Rojas","doi":"10.1007/s00153-023-00886-2","DOIUrl":"10.1007/s00153-023-00886-2","url":null,"abstract":"<div><p>We expand our effective framework for weak convergence of measures on the real line by showing that effective convergence in the Prokhorov metric is equivalent to effective weak convergence. In addition, we establish a framework for the study of the effective theory of vague convergence of measures. We introduce a uniform notion and a non-uniform notion of vague convergence, and we show that both these notions are equivalent. However, limits under effective vague convergence may not be computable even when they are finite. We give an example of a finite incomputable effective vague limit measure, and we provide a necessary and sufficient condition so that effective vague convergence produces a computable limit. Finally, we determine a sufficient condition for which effective weak and vague convergence of measures coincide. As a corollary, we obtain an effective version of the equivalence between classical weak and vague convergence of sequences of probability measures.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"63 1-2","pages":"225 - 238"},"PeriodicalIF":0.3,"publicationDate":"2023-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00153-023-00886-2.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135476054","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
Cut elimination by unthreading 拧开螺纹消除
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2023-09-22 DOI: 10.1007/s00153-023-00892-4
Gabriele Pulcini
{"title":"Cut elimination by unthreading","authors":"Gabriele Pulcini","doi":"10.1007/s00153-023-00892-4","DOIUrl":"10.1007/s00153-023-00892-4","url":null,"abstract":"<div><p>We provide a non-Gentzen, though fully syntactical, cut-elimination algorithm for classical propositional logic. The designed procedure is implemented on <span>(textsf{GS4})</span>, the one-sided version of Kleene’s sequent system <span>(textsf{G4})</span>. The algorithm here proposed proves to be more ‘dexterous’ than other, more traditional, Gentzen-style techniques as the size of proofs decreases at each step of reduction. As a corollary result, we show that analyticity always guarantees minimality of the size of <span>(textsf{GS4})</span>-proofs.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"63 1-2","pages":"211 - 223"},"PeriodicalIF":0.3,"publicationDate":"2023-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00153-023-00892-4.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136060753","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
Superrosiness and dense pairs of geometric structures 几何结构的超级性和密集对
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2023-09-19 DOI: 10.1007/s00153-023-00890-6
Gareth J. Boxall
{"title":"Superrosiness and dense pairs of geometric structures","authors":"Gareth J. Boxall","doi":"10.1007/s00153-023-00890-6","DOIUrl":"10.1007/s00153-023-00890-6","url":null,"abstract":"<div><p>Let <i>T</i> be a complete geometric theory and let <span>(T_P)</span> be the theory of dense pairs of models of <i>T</i>. We show that if <i>T</i> is superrosy with <img>-rank 1 then <span>(T_P)</span> is superrosy with <img>-rank at most <span>(omega )</span>.\u0000</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"63 1-2","pages":"203 - 209"},"PeriodicalIF":0.3,"publicationDate":"2023-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00153-023-00890-6.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135060655","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
Computable approximations of a chainable continuum with a computable endpoint 具有可计算端点的可链连续体的可计算近似值
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2023-09-13 DOI: 10.1007/s00153-023-00891-5
Zvonko Iljazović, Matea Jelić
{"title":"Computable approximations of a chainable continuum with a computable endpoint","authors":"Zvonko Iljazović,&nbsp;Matea Jelić","doi":"10.1007/s00153-023-00891-5","DOIUrl":"10.1007/s00153-023-00891-5","url":null,"abstract":"<div><p>It is known that a semicomputable continuum <i>S</i> in a computable topological space can be approximated by a computable subcontinuum by any given precision under condition that <i>S</i> is chainable and decomposable. In this paper we show that decomposability can be replaced by the assumption that <i>S</i> is chainable from <i>a</i> to <i>b</i>, where <i>a</i> is a computable point.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"63 1-2","pages":"181 - 201"},"PeriodicalIF":0.3,"publicationDate":"2023-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135734368","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
Semi-honest subrecursive degrees and the collection rule in arithmetic 半诚实子递归度与算术中的集合规则
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2023-08-12 DOI: 10.1007/s00153-023-00889-z
Andrés Cordón-Franco, F. Félix Lara-Martín
{"title":"Semi-honest subrecursive degrees and the collection rule in arithmetic","authors":"Andrés Cordón-Franco,&nbsp;F. Félix Lara-Martín","doi":"10.1007/s00153-023-00889-z","DOIUrl":"10.1007/s00153-023-00889-z","url":null,"abstract":"<div><p>By a result of L.D. Beklemishev, the hierarchy of nested applications of the <span>(Sigma _1)</span>-collection rule over any <span>(Pi _2)</span>-axiomatizable base theory extending Elementary Arithmetic collapses to its first level. We prove that this result cannot in general be extended to base theories of arbitrary quantifier complexity. In fact, given any recursively enumerable set of true <span>(Pi _2)</span>-sentences, <i>S</i>, we construct a sound <span>((Sigma _2 ! vee ! Pi _2))</span>-axiomatized theory <i>T</i> extending <i>S</i> such that the hierarchy of nested applications of the <span>(Sigma _1)</span>-collection rule over <i>T</i> is proper. Our construction uses some results on subrecursive degree theory obtained by L. Kristiansen.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"63 1-2","pages":"163 - 180"},"PeriodicalIF":0.3,"publicationDate":"2023-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44525200","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
Convergence of measures after adding a real 增加一个实数后，测量结果趋于一致。
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2023-08-11 DOI: 10.1007/s00153-023-00888-0
Damian Sobota, Lyubomyr Zdomskyy
{"title":"Convergence of measures after adding a real","authors":"Damian Sobota,&nbsp;Lyubomyr Zdomskyy","doi":"10.1007/s00153-023-00888-0","DOIUrl":"10.1007/s00153-023-00888-0","url":null,"abstract":"<div><p>We prove that if <span>(mathcal {A})</span> is an infinite Boolean algebra in the ground model <i>V</i> and <span>(mathbb {P})</span> is a notion of forcing adding any of the following reals: a Cohen real, an unsplit real, or a random real, then, in any <span>(mathbb {P})</span>-generic extension <i>V</i>[<i>G</i>], <span>(mathcal {A})</span> has neither the Nikodym property nor the Grothendieck property. A similar result is also proved for a dominating real and the Nikodym property.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"63 1-2","pages":"135 - 162"},"PeriodicalIF":0.3,"publicationDate":"2023-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10787011/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"52099468","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
A Mathias criterion for the Magidor iteration of Prikry forcings Prikry强迫Magidor迭代的一个Mathias准则
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2023-08-04 DOI: 10.1007/s00153-023-00887-1
Omer Ben-Neria
{"title":"A Mathias criterion for the Magidor iteration of Prikry forcings","authors":"Omer Ben-Neria","doi":"10.1007/s00153-023-00887-1","DOIUrl":"10.1007/s00153-023-00887-1","url":null,"abstract":"<div><p>We prove a Mathias-type criterion for the Magidor iteration of Prikry forcings.\u0000</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"63 1-2","pages":"119 - 134"},"PeriodicalIF":0.3,"publicationDate":"2023-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41471612","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
Herbrand complexity and the epsilon calculus with equality Herbrand复杂度和相等的微积分
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2023-07-29 DOI: 10.1007/s00153-023-00877-3
Kenji Miyamoto, Georg Moser
{"title":"Herbrand complexity and the epsilon calculus with equality","authors":"Kenji Miyamoto,&nbsp;Georg Moser","doi":"10.1007/s00153-023-00877-3","DOIUrl":"10.1007/s00153-023-00877-3","url":null,"abstract":"<div><p>The <span>(varepsilon )</span>-elimination method of Hilbert’s <span>(varepsilon )</span>-calculus yields the up-to-date most direct algorithm for computing the Herbrand disjunction of an extensional formula. A central advantage is that the upper bound on the Herbrand complexity obtained is independent of the propositional structure of the proof. Prior (modern) work on Hilbert’s <span>(varepsilon )</span>-calculus focused mainly on the pure calculus, without equality. We clarify that this independence also holds for first-order logic with equality. Further, we provide upper bounds analyses of the extended first <span>(varepsilon )</span>-theorem, even if the formalisation incorporates so-called <span>(varepsilon )</span>-equality axioms.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"63 1-2","pages":"89 - 118"},"PeriodicalIF":0.3,"publicationDate":"2023-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44538500","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
Revisiting the conservativity of fixpoints over intuitionistic arithmetic 直觉算术上不动点的保守性修正
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2023-07-28 DOI: 10.1007/s00153-023-00878-2
Mattias Granberg Olsson, Graham E. Leigh
{"title":"Revisiting the conservativity of fixpoints over intuitionistic arithmetic","authors":"Mattias Granberg Olsson,&nbsp;Graham E. Leigh","doi":"10.1007/s00153-023-00878-2","DOIUrl":"10.1007/s00153-023-00878-2","url":null,"abstract":"<div><p>This paper presents a novel proof of the conservativity of the intuitionistic theory of strictly positive fixpoints, <span>(widehat{{textrm{ID}}}{}_{1}^{{textrm{i}}}{})</span>, over Heyting arithmetic (<span>({textrm{HA}})</span>), originally proved in full generality by Arai (Ann Pure Appl Log 162:807–815, 2011. https://doi.org/10.1016/j.apal.2011.03.002). The proof embeds <span>(widehat{{textrm{ID}}}{}_{1}^{{textrm{i}}}{})</span> into the corresponding theory over Beeson’s logic of partial terms and then uses two consecutive interpretations, a realizability interpretation of this theory into the subtheory generated by almost negative fixpoints, and a direct interpretation into Heyting arithmetic with partial terms using a hierarchy of satisfaction predicates for almost negative formulae. It concludes by applying van den Berg and van Slooten’s result (Indag Math 29:260–275, 2018. https://doi.org/10.1016/j.indag.2017.07.009) that Heyting arithmetic with partial terms plus the schema of self realizability for arithmetic formulae is conservative over <span>({textrm{HA}})</span>.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"63 1-2","pages":"61 - 87"},"PeriodicalIF":0.3,"publicationDate":"2023-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00153-023-00878-2.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47599322","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
Turing degrees and randomness for continuous measures 连续测度的图灵度与随机性
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2023-06-15 DOI: 10.1007/s00153-023-00873-7
Mingyang Li, Jan Reimann
{"title":"Turing degrees and randomness for continuous measures","authors":"Mingyang Li,&nbsp;Jan Reimann","doi":"10.1007/s00153-023-00873-7","DOIUrl":"10.1007/s00153-023-00873-7","url":null,"abstract":"<div><p>We study degree-theoretic properties of reals that are not random with respect to any continuous probability measure (NCR). To this end, we introduce a family of generalized Hausdorff measures based on the iterates of the “dissipation” function of a continuous measure and study the effective nullsets given by the corresponding Solovay tests. We introduce two constructions that preserve non-randomness with respect to a given continuous measure. This enables us to prove the existence of NCR reals in a number of Turing degrees. In particular, we show that every <span>(Delta ^0_2)</span>-degree contains an NCR element.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"63 1-2","pages":"39 - 59"},"PeriodicalIF":0.3,"publicationDate":"2023-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49534997","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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