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On diagonal functions for equivalence relations 论等价关系的对角函数
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2023-10-18 DOI: 10.1007/s00153-023-00896-0
Serikzhan A. Badaev, Nikolay A. Bazhenov, Birzhan S. Kalmurzayev, Manat Mustafa
{"title":"On diagonal functions for equivalence relations","authors":"Serikzhan A. Badaev,&nbsp;Nikolay A. Bazhenov,&nbsp;Birzhan S. Kalmurzayev,&nbsp;Manat Mustafa","doi":"10.1007/s00153-023-00896-0","DOIUrl":"10.1007/s00153-023-00896-0","url":null,"abstract":"<div><p>We work with weakly precomplete equivalence relations introduced by Badaev. The weak precompleteness is a natural notion inspired by various fixed point theorems in computability theory. Let <i>E</i> be an equivalence relation on the set of natural numbers <span>(omega )</span>, having at least two classes. A total function <i>f</i> is a <i>diagonal function</i> for <i>E</i> if for every <i>x</i>, the numbers <i>x</i> and <i>f</i>(<i>x</i>) are not <i>E</i>-equivalent. It is known that in the case of c.e. relations <i>E</i>, the weak precompleteness of <i>E</i> is equivalent to the lack of computable diagonal functions for <i>E</i>. Here we prove that this result fails already for <span>(Delta ^0_2)</span> equivalence relations, starting with the <span>(Pi ^{-1}_2)</span> level. We focus on the Turing degrees of possible diagonal functions. We prove that for any noncomputable c.e. degree <span>({textbf{d}})</span>, there exists a weakly precomplete c.e. equivalence <i>E</i> admitting a <span>({textbf{d}})</span>-computable diagonal function. We observe that a Turing degree <span>({textbf{d}})</span> can compute a diagonal function for every <span>(Delta ^0_2)</span> equivalence relation <i>E</i> if and only if <span>({textbf{d}})</span> computes <span>({textbf{0}}')</span>. On the other hand, every PA degree can compute a diagonal function for an arbitrary c.e. equivalence <i>E</i>. In addition, if <span>({textbf{d}})</span> computes diagonal functions for all c.e. <i>E</i>, then <span>({textbf{d}})</span> must be a DNC degree.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"63 3-4","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00153-023-00896-0.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135884718","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
Compositional truth with propositional tautologies and quantifier-free correctness 带有命题同义反复和无量词正确性的组合真理
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2023-10-06 DOI: 10.1007/s00153-023-00893-3
Bartosz Wcisło
{"title":"Compositional truth with propositional tautologies and quantifier-free correctness","authors":"Bartosz Wcisło","doi":"10.1007/s00153-023-00893-3","DOIUrl":"10.1007/s00153-023-00893-3","url":null,"abstract":"<div><p>In Cieśliński (J Philos Logic 39:325–337, 2010), Cieśliński asked whether compositional truth theory with the additional axiom that all propositional tautologies are true is conservative over Peano Arithmetic. We provide a partial answer to this question, showing that if we additionally assume that truth predicate agrees with arithmetical truth on quantifier-free sentences, the resulting theory is as strong as <span>(Delta _0)</span>-induction for the compositional truth predicate, hence non-conservative. On the other hand, it can be shown with a routine argument that the principle of quantifier-free correctness is itself conservative.\u0000</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"63 1-2","pages":"239 - 257"},"PeriodicalIF":0.3,"publicationDate":"2023-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00153-023-00893-3.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135302156","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
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
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