Annals of Pure and Applied Logic最新文献

{"title":"On the spectrum of limit models","authors":"Jeremy Beard ,&nbsp;Marcos Mazari-Armida","doi":"10.1016/j.apal.2025.103647","DOIUrl":"10.1016/j.apal.2025.103647","url":null,"abstract":"<div><div>We study the spectrum of limit models assuming the existence of a nicely behaved independence notion. Under reasonable assumptions, we show that all ‘long’ limit models are isomorphic, and all ‘short’ limit models are non-isomorphic. <section><p><strong>Theorem</strong></p><div><em>Let</em> <strong>K</strong> <em>be a</em> <span><math><msub><mrow><mi>ℵ</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span><em>-tame abstract elementary class stable in</em> <span><math><mi>λ</mi><mo>≥</mo><mi>LS</mi><mo>(</mo><mi>K</mi><mo>)</mo></math></span> <em>with amalgamation, joint embedding and no maximal models. Let</em> <span><math><mi>κ</mi><mo>&lt;</mo><msup><mrow><mi>λ</mi></mrow><mrow><mo>+</mo></mrow></msup></math></span> <em>be regular. Suppose</em> <figure><img></figure> <em>is an independence relation on the models of size λ that satisfies uniqueness, extension, non-forking amalgamation, universal continuity, and</em> <span><math><mo>(</mo><mo>≥</mo><mi>κ</mi><mo>)</mo></math></span><em>-local character.</em></div><div><em>Suppose</em> <span><math><msub><mrow><mi>δ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>δ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>&lt;</mo><msup><mrow><mi>λ</mi></mrow><mrow><mo>+</mo></mrow></msup></math></span> <em>with</em> <span><math><mi>cf</mi><mo>(</mo><msub><mrow><mi>δ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>&lt;</mo><mi>cf</mi><mo>(</mo><msub><mrow><mi>δ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span><em>. Then for any</em> <span><math><msub><mrow><mi>N</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>N</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mi>M</mi><mo>∈</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>λ</mi></mrow></msub></math></span> <em>where</em> <span><math><msub><mrow><mi>N</mi></mrow><mrow><mi>l</mi></mrow></msub></math></span> <em>is a</em> <span><math><mo>(</mo><mi>λ</mi><mo>,</mo><msub><mrow><mi>δ</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>)</mo></math></span><em>-limit model over M for</em> <span><math><mi>l</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn></math></span><em>,</em><span><span><img></span></span></div></section></div><div>Both implications in the conclusion have improvements. High cofinality limits are isomorphic without the <span><math><msub><mrow><mi>ℵ</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>-tameness assumption and assuming <figure><img></figure> is defined only on high cofinality limit models. Low cofinality limits are non-isomorphic without assuming non-forking amalgamation.</div><div>We show how our results can be used to study limit models in both abstract settings and in natural examples of abstract elementary classes.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 10","pages":"Article 103647"},"PeriodicalIF":0.6,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144902348","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 Recurrence Axioms","authors":"Sakaé Fuchino ,&nbsp;Toshimichi Usuba","doi":"10.1016/j.apal.2025.103631","DOIUrl":"10.1016/j.apal.2025.103631","url":null,"abstract":"<div><div>The Recurrence Axiom for a class <span><math><mi>P</mi></math></span> of posets and a set <em>A</em> of parameters is an axiom scheme in the language of <span>ZFC</span> asserting that if a statement with parameters from <em>A</em> is forced by a poset in <span><math><mi>P</mi></math></span>, then there is a ground containing the parameters and satisfying the statement.</div><div>The tightly super-<span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>(</mo><mo>∞</mo><mo>)</mo></mrow></msup></math></span>-<span><math><mi>P</mi></math></span>-Laver generic hyperhuge continuum implies the Recurrence Axiom for <span><math><mi>P</mi></math></span> and <span><math><mi>H</mi><mo>(</mo><msup><mrow><mn>2</mn></mrow><mrow><msub><mrow><mi>ℵ</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></msup><mo>)</mo></math></span>. The consistency strength of this assumption can be decided thanks to our main theorems asserting that the minimal ground (bedrock) exists under a tightly <span><math><mi>P</mi></math></span>-generic hyperhuge cardinal <em>κ</em>, and that <em>κ</em> in the bedrock is genuinely hyperhuge, or even super <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>(</mo><mo>∞</mo><mo>)</mo></mrow></msup></math></span> hyperhuge if <em>κ</em> is a tightly super-<span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>(</mo><mo>∞</mo><mo>)</mo></mrow></msup></math></span>-<span><math><mi>P</mi></math></span>-Laver generic hyperhuge definable cardinal.</div><div>The Laver Generic Maximum (<span>LGM</span>), one of the strongest combinations of axioms in our context, integrates practically all known set-theoretic principles and axioms in itself, either as its consequences or as theorems holding in (many) grounds of the universe. For instance, double plus version of Martin's Maximum is a consequence of <span>LGM</span> while Cichoń's Maximum is a phenomenon in many grounds of the universe under <span>LGM</span>.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 10","pages":"Article 103631"},"PeriodicalIF":0.6,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144556730","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":"Transposition of variables is hard to axiomatize","authors":"Hajnal Andréka,&nbsp;István Németi,&nbsp;Zsolt Tuza","doi":"10.1016/j.apal.2025.103650","DOIUrl":"10.1016/j.apal.2025.103650","url":null,"abstract":"<div><div>The function <span><math><msub><mrow><mi>p</mi></mrow><mrow><mi>x</mi><mi>y</mi></mrow></msub></math></span> that interchanges two logical variables <span><math><mi>x</mi><mo>,</mo><mi>y</mi></math></span> in formulas is hard to describe in the following sense. Let <em>F</em> denote the Lindenbaum–Tarski formula-algebra of a finite-variable first-order logic, endowed with <span><math><msub><mrow><mi>p</mi></mrow><mrow><mi>x</mi><mi>y</mi></mrow></msub></math></span> as a unary function. We prove that each equational axiom system for the equational theory of <em>F</em> has to contain, for each finite <em>n</em>, an equation that contains together with <span><math><msub><mrow><mi>p</mi></mrow><mrow><mi>x</mi><mi>y</mi></mrow></msub></math></span> at least <em>n</em> algebraic variables, and each of the operations <span><math><mo>∃</mo><mo>,</mo><mo>=</mo><mo>,</mo><mo>∨</mo></math></span>. This gives an answer to a problem raised by Johnson (1969) <span><span>[30]</span></span>: the class <span><math><mi>R</mi><mi>P</mi><mi>E</mi><msub><mrow><mi>A</mi></mrow><mrow><mi>α</mi></mrow></msub></math></span> of representable polyadic equality algebras of a finite dimension <span><math><mi>α</mi><mo>≥</mo><mn>3</mn></math></span> cannot be axiomatized by adding finitely many equations to the equational theory of representable cylindric algebras of dimension <em>α</em>. Consequences for proof systems of finite-variable logic and for defining equations of polyadic equality algebras are given.</div><div>The proof uses a family of nonrepresentable polyadic equality algebras <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> that are more and more nearly representable as <em>n</em> increases: their <em>n</em>-generated subalgebras as well as their proper reducts are representable. The lattice of subvarieties of <span><math><mi>R</mi><mi>P</mi><mi>E</mi><msub><mrow><mi>A</mi></mrow><mrow><mi>α</mi></mrow></msub></math></span> is investigated and new open problems are asked about the interplay between the transposition operations and about generalizability of the results to infinite dimensions.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 10","pages":"Article 103650"},"PeriodicalIF":0.6,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144906840","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":"An explicit Kuznetsov-Muravitsky enrichment","authors":"Mamuka Jibladze,&nbsp;Evgeny Kuznetsov","doi":"10.1016/j.apal.2025.103628","DOIUrl":"10.1016/j.apal.2025.103628","url":null,"abstract":"<div><div>An embedding of arbitrary Heyting algebra <em>H</em> into a reduct from the variety of Kuznetsov-Muravitsky algebras is constructed. An algebraic proof is given that this reduct belongs to the variety of Heyting algebras generated by <em>H</em>.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 10","pages":"Article 103628"},"PeriodicalIF":0.6,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144364463","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":"Topological games in Ramsey spaces","authors":"Julián C. Cano ,&nbsp;Carlos A. Di Prisco","doi":"10.1016/j.apal.2025.103630","DOIUrl":"10.1016/j.apal.2025.103630","url":null,"abstract":"<div><div>Topological Ramsey theory studies a class of combinatorial topological spaces, known as topological Ramsey spaces, unifying the essential features of those combinatorial frames where the Ramsey property is equivalent to the Baire property. In this article, we present a general overview of the combinatorial structure of topological Ramsey spaces and their main properties, and we propose an alternative proof of the abstract Ellentuck theorem for a large family of axiomatized topological Ramsey spaces. Additionally, we introduce the notion of selective axiomatized topological Ramsey space, and generalize Kastanas games in order to characterize the Ramsey property for this broad family of topological Ramsey spaces through topological games.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 10","pages":"Article 103630"},"PeriodicalIF":0.6,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144514036","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 definable Skolem functions and trichotomy","authors":"Bruno Dinis ,&nbsp;Mário J. Edmundo","doi":"10.1016/j.apal.2025.103632","DOIUrl":"10.1016/j.apal.2025.103632","url":null,"abstract":"<div><div>In this paper we give an explicit characterization of o-minimal structures with definable Skolem functions/definable choice. Such structures are, after naming finitely many elements from the prime model, a union of finitely many trivial points each defined over ∅ and finitely many open intervals each a union of a ∅-definable family of group-intervals with fixed positive elements.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 10","pages":"Article 103632"},"PeriodicalIF":0.6,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144580943","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":"Unstable independence from the categorical point of view","authors":"Mark Kamsma ,&nbsp;Jiří Rosický","doi":"10.1016/j.apal.2025.103629","DOIUrl":"10.1016/j.apal.2025.103629","url":null,"abstract":"<div><div>We give a category-theoretic construction of simple and NSOP₁-like independence relations in locally finitely presentable categories, and in the more general locally finitely multipresentable categories. We do so by identifying properties of a class of monomorphisms <span><math><mi>M</mi></math></span> such that the pullback squares consisting of morphisms in <span><math><mi>M</mi></math></span> form the desired independence relation. This generalizes the category-theoretic construction of stable independence relations using effective unions or cellular squares by M. Lieberman, S. Vasey and the second author to the unstable setting.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 10","pages":"Article 103629"},"PeriodicalIF":0.6,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144322319","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":"The unstable formula theorem revisited via algorithms","authors":"M. Malliaris ,&nbsp;S. Moran","doi":"10.1016/j.apal.2025.103633","DOIUrl":"10.1016/j.apal.2025.103633","url":null,"abstract":"<div><div>This paper is about the surprising interaction of a foundational result from model theory, about stability of theories, with algorithmic stability in learning. First, in response to gaps in existing learning models, we introduce a new statistical learning model, called “Probably Eventually Correct” or PEC. We characterize Littlestone (stable) classes in terms of this model. As a corollary, Littlestone classes have frequent short definitions in a natural statistical sense. In order to obtain a characterization of Littlestone classes in terms of frequent definitions, we build an equivalence theorem highlighting what is common to many existing approximation algorithms, and to the new PEC. This is guided by an analogy to definability of types in model theory, but has its own character. Drawing on these theorems and on other recent work, we present a complete algorithmic analogue of Shelah's celebrated Unstable Formula Theorem, with algorithmic properties taking the place of the infinite.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 10","pages":"Article 103633"},"PeriodicalIF":0.6,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144632237","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":"Automorphisms of P(ω)/fin and large continuum","authors":"Alan Dow","doi":"10.1016/j.apal.2025.103627","DOIUrl":"10.1016/j.apal.2025.103627","url":null,"abstract":"<div><div>We prove that it is consistent with <span><math><mi>c</mi><mo>&gt;</mo><msub><mrow><mi>ℵ</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> that all automorphisms of <span><math><mi>P</mi><mo>(</mo><mi>ω</mi><mo>)</mo><mo>/</mo><mtext>fin</mtext></math></span> are trivial.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 10","pages":"Article 103627"},"PeriodicalIF":0.6,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144263932","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":"The DeMorganization of a locale","authors":"Igor Arrieta","doi":"10.1016/j.apal.2025.103634","DOIUrl":"10.1016/j.apal.2025.103634","url":null,"abstract":"<div><div>In 2009, Caramello proved that each topos has a largest dense subtopos whose internal logic satisfies De Morgan law (also known as the law of the weak excluded middle). This finding implies that every locale has a largest dense extremally disconnected sublocale, referred to as its DeMorganization. In this paper, we take the first steps in exploring the DeMorganization in the localic context, shedding light on its geometric nature by showing that it is always a fitted sublocale and by providing a concrete description. Explicit examples of DeMorganizations for toposes that do not satisfy De Morgan law are rather difficult to find. We present a contribution in that direction, with the main result of the paper showing that for any metrizable locale (without isolated points), its DeMorganization coincides with its Booleanization. This, in particular, implies that any extremally disconnected metric locale (without isolated points) must be Boolean, generalizing a well-known result for topological spaces to the localic setting.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 10","pages":"Article 103634"},"PeriodicalIF":0.6,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144587704","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}