Annals of Pure and Applied Logic最新文献

{"title":"The formal verification of the ctm approach to forcing","authors":"Emmanuel Gunther ,&nbsp;Miguel Pagano ,&nbsp;Pedro Sánchez Terraf ,&nbsp;Matías Steinberg","doi":"10.1016/j.apal.2024.103413","DOIUrl":"10.1016/j.apal.2024.103413","url":null,"abstract":"<div><p>We discuss some highlights of our computer-verified proof of the construction, given a countable transitive set-model <em>M</em> of <em>ZFC</em>, of generic extensions satisfying <span><math><mrow><mi>ZFC</mi></mrow><mo>+</mo><mo>¬</mo><mrow><mi>CH</mi></mrow></math></span> and <span><math><mrow><mi>ZFC</mi></mrow><mo>+</mo><mrow><mi>CH</mi></mrow></math></span>. Moreover, let <span><math><mi>R</mi></math></span> be the set of instances of the Axiom of Replacement. We isolated a 21-element subset <span><math><mi>Ω</mi><mo>⊆</mo><mi>R</mi></math></span> and defined <span><math><mi>F</mi><mo>:</mo><mi>R</mi><mo>→</mo><mi>R</mi></math></span> such that for every <span><math><mi>Φ</mi><mo>⊆</mo><mi>R</mi></math></span> and <em>M</em>-generic <em>G</em>, <span><math><mi>M</mi><mo>⊨</mo><mrow><mi>ZC</mi></mrow><mo>∪</mo><mi>F</mi><mtext>“</mtext><mi>Φ</mi><mo>∪</mo><mi>Ω</mi></math></span> implies <span><math><mi>M</mi><mo>[</mo><mi>G</mi><mo>]</mo><mo>⊨</mo><mrow><mi>ZC</mi></mrow><mo>∪</mo><mi>Φ</mi><mo>∪</mo><mo>{</mo><mo>¬</mo><mrow><mi>CH</mi></mrow><mo>}</mo></math></span>, where <em>ZC</em> is Zermelo set theory with Choice.</p><p>To achieve this, we worked in the proof assistant <em>Isabelle</em>, basing our development on the Isabelle/ZF library by L. Paulson and others.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"175 5","pages":"Article 103413"},"PeriodicalIF":0.8,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139649384","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Sharp Vaught's conjecture for some classes of partial orders","authors":"Miloš S. Kurilić","doi":"10.1016/j.apal.2024.103411","DOIUrl":"10.1016/j.apal.2024.103411","url":null,"abstract":"<div><p>Matatyahu Rubin has shown that a sharp version of Vaught's conjecture, <span><math><mi>I</mi><mo>(</mo><mi>T</mi><mo>,</mo><mi>ω</mi><mo>)</mo><mo>∈</mo><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mi>c</mi><mo>}</mo></math></span>, holds for each complete theory of linear order <span><math><mi>T</mi></math></span>. We show that the same is true for each complete theory of partial order having a model in the minimal class of partial orders containing the class of linear orders and which is closed under finite products and finite disjoint unions. The same holds for the extension of the class of rooted trees admitting a finite monomorphic decomposition, obtained in the same way. The sharp version of Vaught's conjecture also holds for the theories of trees which are infinite disjoint unions of linear orders.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"175 4","pages":"Article 103411"},"PeriodicalIF":0.8,"publicationDate":"2024-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139462598","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Towards characterizing the >ω2-fickle recursively enumerable Turing degrees","authors":"Liling Ko","doi":"10.1016/j.apal.2023.103403","DOIUrl":"10.1016/j.apal.2023.103403","url":null,"abstract":"<div><p><span>Given a finite lattice </span><em>L</em><span> that can be embedded in the recursively enumerable (r.e.) Turing degrees </span><span><math><mo>〈</mo><msub><mrow><mi>R</mi></mrow><mrow><mi>T</mi></mrow></msub><mo>,</mo><msub><mrow><mo>≤</mo></mrow><mrow><mi>T</mi></mrow></msub><mo>〉</mo></math></span>, it is not known how one can characterize the degrees <span><math><mi>d</mi><mo>∈</mo><msub><mrow><mi>R</mi></mrow><mrow><mi>T</mi></mrow></msub></math></span> below which <em>L</em> can be embedded. Two important characterizations are of the <span><math><msub><mrow><mi>L</mi></mrow><mrow><mn>7</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>M</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span> lattices, where the lattices are embedded below <strong>d</strong> if and only if <strong>d</strong> contains sets of “<em>fickleness</em>” &gt;<em>ω</em> and <span><math><mo>≥</mo><msup><mrow><mi>ω</mi></mrow><mrow><mi>ω</mi></mrow></msup></math></span> respectively. We work towards finding a lattice that characterizes the levels above <span><math><msup><mrow><mi>ω</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>, the first non-trivial level after <em>ω</em>. We considered lattices that are as “short” in height and “narrow” in width as <span><math><msub><mrow><mi>L</mi></mrow><mrow><mn>7</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>M</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span>, but the lattices characterize also the &gt;<em>ω</em> or <span><math><mo>≥</mo><msup><mrow><mi>ω</mi></mrow><mrow><mi>ω</mi></mrow></msup></math></span> levels, if the lattices are not already embeddable below all non-zero r.e. degrees. We also considered upper semilattices (USLs) by removing the bottom meet(s) of some previously considered lattices, but the removals did not change the levels characterized. We discovered three lattices besides <span><math><msub><mrow><mi>M</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span> that also characterize the <span><math><mo>≥</mo><msup><mrow><mi>ω</mi></mrow><mrow><mi>ω</mi></mrow></msup></math></span>-levels. Our search for <span><math><mo>&gt;</mo><msup><mrow><mi>ω</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-candidates can therefore be reduced to the lattice-theoretic problem of finding lattices that do not contain any of the four <span><math><mo>≥</mo><msup><mrow><mi>ω</mi></mrow><mrow><mi>ω</mi></mrow></msup></math></span><span>-lattices as sublattices.</span></p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"175 4","pages":"Article 103403"},"PeriodicalIF":0.8,"publicationDate":"2024-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139104768","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Structural and universal completeness in algebra and logic","authors":"Paolo Aglianò ,&nbsp;Sara Ugolini","doi":"10.1016/j.apal.2023.103391","DOIUrl":"10.1016/j.apal.2023.103391","url":null,"abstract":"<div><p>In this work we study the notions of structural and universal completeness both from the algebraic and logical point of view. In particular, we provide new algebraic characterizations of quasivarieties that are actively and passively universally complete, and passively structurally complete. We apply these general results to varieties of bounded lattices and to quasivarieties related to substructural logics. In particular we show that a substructural logic satisfying weakening is passively structurally complete if and only if every classical contradiction is explosive in it. Moreover, we fully characterize the passively structurally complete varieties of <span><math><mi>MTL</mi></math></span>-algebras, i.e., bounded commutative integral residuated lattices generated by chains.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"175 3","pages":"Article 103391"},"PeriodicalIF":0.8,"publicationDate":"2023-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0168007223001483/pdfft?md5=566d39a3ec46ed86a39bb7490723fd1b&pid=1-s2.0-S0168007223001483-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138683789","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"SOP1, SOP2, and antichain tree property","authors":"JinHoo Ahn ,&nbsp;Joonhee Kim","doi":"10.1016/j.apal.2023.103402","DOIUrl":"10.1016/j.apal.2023.103402","url":null,"abstract":"<div><p>In this paper, we study some tree properties and their related indiscernibilities. First, we prove that SOP₂ can be witnessed by a formula with a tree of tuples holding ‘arbitrary homogeneous inconsistency’ (e.g., weak <em>k</em>-TP₁ conditions or other possible inconsistency configurations).</p><p>And we introduce a notion of tree-indiscernibility, which preserves witnesses of SOP₁, and by using this, we investigate the problem of (in)equality of SOP₁ and SOP₂.</p><p>Assuming the existence of a formula having SOP₁ such that no finite conjunction of it has SOP₂<span>, we observe that the formula must witness some tree-property-like phenomenon, which we will call the antichain tree property (ATP, see </span><span>Definition 4.1</span>). We show that ATP implies SOP₁ and TP₂, but the converse of each implication does not hold. So the class of NATP theories (theories without ATP) contains the class of NSOP₁ theories and the class of NTP₂ theories.</p><p>At the end of the paper, we construct a structure whose theory has a formula having ATP, but any conjunction of the formula does not have SOP₂. So this example shows that SOP₁ and SOP₂ are not the same at the level of formulas, i.e., there is a formula having SOP₁, while any finite conjunction of it does not witness SOP₂ (but a variation of the formula still has SOP₂).</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"175 3","pages":"Article 103402"},"PeriodicalIF":0.8,"publicationDate":"2023-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138575042","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Constructing the constructible universe constructively","authors":"Richard Matthews ,&nbsp;Michael Rathjen","doi":"10.1016/j.apal.2023.103392","DOIUrl":"10.1016/j.apal.2023.103392","url":null,"abstract":"<div><p>We study the properties of the constructible universe, L, over intuitionistic theories. We give an extended set of fundamental operations which is sufficient to generate the universe over Intuitionistic Kripke-Platek set theory without Infinity. Following this, we investigate when L can fail to be an inner model in the traditional sense. Namely, we show that over Constructive Zermelo-Fraenkel (even with the Power Set axiom) one cannot prove that the Axiom of Exponentiation holds in L.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"175 3","pages":"Article 103392"},"PeriodicalIF":0.8,"publicationDate":"2023-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138548019","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On categorical structures arising from implicative algebras: From topology to assemblies","authors":"Samuele Maschio,&nbsp;Davide Trotta","doi":"10.1016/j.apal.2023.103390","DOIUrl":"https://doi.org/10.1016/j.apal.2023.103390","url":null,"abstract":"<div><p>Implicative algebras have been recently introduced by Miquel in order to provide a unifying notion of model, encompassing the most relevant and used ones, such as realizability (both classical and intuitionistic), and forcing. In this work, we initially approach implicative algebras as a generalization of locales, and we extend several topological-like concepts to the realm of implicative algebras, accompanied by various concrete examples. Then, we shift our focus to viewing implicative algebras as a generalization of partial combinatory algebras. We abstract the notion of a category of assemblies, partition assemblies, and modest sets to arbitrary implicative algebras, and thoroughly investigate their categorical properties and interrelationships.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"175 3","pages":"Article 103390"},"PeriodicalIF":0.8,"publicationDate":"2023-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0168007223001471/pdfft?md5=e62dfd1384a82cf70c38bd805eb49847&pid=1-s2.0-S0168007223001471-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138484884","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Probabilistic temporal logic with countably additive semantics","authors":"Dragan Doder ,&nbsp;Zoran Ognjanović","doi":"10.1016/j.apal.2023.103389","DOIUrl":"10.1016/j.apal.2023.103389","url":null,"abstract":"<div><p>This work presents a proof-theoretical and model-theoretical approach to probabilistic temporal logic. We present two novel logics; each of them extends both the language of linear time logic (LTL) and the language of probabilistic logic with polynomial weight formulas. The first logic is designed for reasoning about probabilities of temporal events, allowing statements like “the probability that A will hold in next moment is at least the probability that B will always hold” and conditional probability statements like “probability that A will always hold, given that B holds, is at least one half”, where A and B are arbitrary statements. We axiomatize this logic, provide corresponding sigma additive semantics and prove that the axiomatization is sound and strongly complete. We show that the satisfiability problem for our logic is decidable, by presenting a procedure which runs in polynomial space. We also present a logic with much richer language, in which probabilities are not attached only to temporal events, but the language allows arbitrary nesting of probability and temporal operators, allowing statements like “probability that tomorrow the chance of rain will be less than 80% is at least a half”. For this logic we prove a decidability result.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"175 9","pages":"Article 103389"},"PeriodicalIF":0.8,"publicationDate":"2023-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S016800722300146X/pdfft?md5=b48bdc121fa115161627545db4c861fd&pid=1-s2.0-S016800722300146X-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135716247","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On duality and model theory for polyadic spaces","authors":"Sam van Gool,&nbsp;Jérémie Marquès","doi":"10.1016/j.apal.2023.103388","DOIUrl":"https://doi.org/10.1016/j.apal.2023.103388","url":null,"abstract":"<div><p>This paper is a study of first-order coherent logic from the point of view of duality and categorical logic. We prove a duality theorem between coherent hyperdoctrines and open polyadic Priestley spaces, which we subsequently apply to prove completeness, omitting types, and Craig interpolation theorems for coherent or intuitionistic logic. Our approach emphasizes the role of interpolation and openness properties, and allows for a modular, syntax-free treatment of these model-theoretic results. As further applications of the same method, we prove completeness theorems for constant domain and Gödel-Dummett intuitionistic predicate logics.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"175 2","pages":"Article 103388"},"PeriodicalIF":0.8,"publicationDate":"2023-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134656886","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Pathologies in satisfaction classes","authors":"Athar Abdul-Quader ,&nbsp;Mateusz Łełyk","doi":"10.1016/j.apal.2023.103387","DOIUrl":"https://doi.org/10.1016/j.apal.2023.103387","url":null,"abstract":"<div><p>We study subsets of countable recursively saturated models of <span><math><mi>PA</mi></math></span> which can be defined using pathologies in satisfaction classes. More precisely, we characterize those subsets <em>X</em> such that there is a satisfaction class <em>S</em> where <em>S</em> behaves correctly on an idempotent disjunction of length <em>c</em> if and only if <span><math><mi>c</mi><mo>∈</mo><mi>X</mi></math></span>. We generalize this result to characterize several types of pathologies including double negations, blocks of extraneous quantifiers, and binary disjunctions and conjunctions. We find a surprising relationship between the cuts which can be defined in this way and arithmetic saturation: namely, a countable nonstandard model is arithmetically saturated if and only if every cut can be the “idempotent disjunctively correct cut” in some satisfaction class. We describe the relationship between types of pathologies and the closure properties of the cuts defined by these pathologies.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"175 2","pages":"Article 103387"},"PeriodicalIF":0.8,"publicationDate":"2023-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49725035","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}