{"title":"Descriptive complexity of topological invariants","authors":"Djamel Eddine Amir, Mathieu Hoyrup","doi":"10.1016/j.apal.2025.103611","DOIUrl":"10.1016/j.apal.2025.103611","url":null,"abstract":"<div><div>In this article, we investigate the descriptive complexity of topological invariants. Our main goal is to understand the expressive power of low complexity invariants, by investigating which spaces they can distinguish. We study the invariants in the first two levels of the Borel hierarchy. We develop techniques to establish whether two spaces can be separated by invariants in these levels. We show that they are sufficient to separate finite topological graphs. We finally identify the complexity of recognizing the line segment.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 8","pages":"Article 103611"},"PeriodicalIF":0.6,"publicationDate":"2025-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144068773","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}
Wesley Calvert , Douglas Cenzer , David Gonzalez , Valentina Harizanov
{"title":"Generically computable linear orderings","authors":"Wesley Calvert , Douglas Cenzer , David Gonzalez , Valentina Harizanov","doi":"10.1016/j.apal.2025.103612","DOIUrl":"10.1016/j.apal.2025.103612","url":null,"abstract":"<div><div>We study notions of generic and coarse computability in the context of computable structure theory. Our notions are stratified by the <span><math><msub><mrow><mi>Σ</mi></mrow><mrow><mi>β</mi></mrow></msub></math></span> hierarchy. We focus on linear orderings. We show that at the <span><math><msub><mrow><mi>Σ</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> level, all linear orderings have both generically and coarsely computable copies. This behavior changes abruptly at higher levels; we show that at the <span><math><msub><mrow><mi>Σ</mi></mrow><mrow><mi>α</mi><mo>+</mo><mn>2</mn></mrow></msub></math></span> level for any <span><math><mi>α</mi><mo>∈</mo><msubsup><mrow><mi>ω</mi></mrow><mrow><mn>1</mn></mrow><mrow><mi>C</mi><mi>K</mi></mrow></msubsup></math></span> the set of linear orderings with generically or coarsely computable copies is <span><math><msubsup><mrow><mi>Σ</mi></mrow><mrow><mn>1</mn></mrow><mrow><mn>1</mn></mrow></msubsup></math></span>-complete and therefore maximally complicated. This development is new even in the general analysis of generic and coarse computability of countable structures. In the process of proving these results, we introduce new tools for understanding generically and coarsely computable structures. We are able to give a purely structural statement that is equivalent to having a generically computable copy and show that every relational structure with only finitely many relations has coarsely and generically computable copies at the lowest level of the hierarchy.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 8","pages":"Article 103612"},"PeriodicalIF":0.6,"publicationDate":"2025-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143943554","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":"Spaces not distinguishing ideal pointwise and σ-uniform convergence","authors":"Rafał Filipów, Adam Kwela","doi":"10.1016/j.apal.2025.103609","DOIUrl":"10.1016/j.apal.2025.103609","url":null,"abstract":"<div><div>We examine topological spaces not distinguishing ideal pointwise and ideal <em>σ</em>-uniform convergence of sequences of real-valued continuous functions defined on them. For instance, we introduce a purely combinatorial cardinal characteristic (a sort of the bounding number <span><math><mi>b</mi></math></span>) and prove that it describes the minimal cardinality of topological spaces which distinguish ideal pointwise and ideal <em>σ</em>-uniform convergence. Moreover, we provide examples of topological spaces (focusing on subsets of reals) that do or do not distinguish the considered convergences. Since similar investigations for ideal quasi-normal convergence instead of ideal <em>σ</em>-uniform convergence have been performed in literature, we also study spaces not distinguishing ideal quasi-normal and ideal <em>σ</em>-uniform convergence of sequences of real-valued continuous functions defined on them.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 9","pages":"Article 103609"},"PeriodicalIF":0.6,"publicationDate":"2025-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143943439","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":"Free p-algebras revisited: An algebraic investigation of implication-free intuitionism","authors":"Tomasz Kowalski , Katarzyna Słomczyńska","doi":"10.1016/j.apal.2025.103610","DOIUrl":"10.1016/j.apal.2025.103610","url":null,"abstract":"<div><div>We give a new construction of free distributive p-algebras. Our construction relies on a detailed description of completely meet-irreducible congruences, so it is purely universal algebraic. It yields a normal form theorem for p-algebra terms, simpler proofs of several existing results. As a by-product, we obtain an isomorphism between the free pseudocomplemented semilattice and the poset of join-irreducibles of the free p-algebra augmented by zero.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 9","pages":"Article 103610"},"PeriodicalIF":0.6,"publicationDate":"2025-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143943440","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 cardinal invariants related to Rosenthal families and large-scale topology","authors":"Arturo Martínez-Celis, Tomasz Żuchowski","doi":"10.1016/j.apal.2025.103607","DOIUrl":"10.1016/j.apal.2025.103607","url":null,"abstract":"<div><div>Given a function <span><math><mi>f</mi><mo>∈</mo><msup><mrow><mi>ω</mi></mrow><mrow><mi>ω</mi></mrow></msup></math></span>, a set <span><math><mi>A</mi><mo>∈</mo><msup><mrow><mo>[</mo><mi>ω</mi><mo>]</mo></mrow><mrow><mi>ω</mi></mrow></msup></math></span> is <em>free for f</em> if <span><math><mi>f</mi><mo>[</mo><mi>A</mi><mo>]</mo><mo>∩</mo><mi>A</mi></math></span> is finite. For a class of functions <span><math><mi>Γ</mi><mo>⊆</mo><msup><mrow><mi>ω</mi></mrow><mrow><mi>ω</mi></mrow></msup></math></span>, we define <span><math><msub><mrow><mi>ros</mi></mrow><mrow><mi>Γ</mi></mrow></msub></math></span> as the smallest size of a family <span><math><mi>A</mi><mo>⊆</mo><msup><mrow><mo>[</mo><mi>ω</mi><mo>]</mo></mrow><mrow><mi>ω</mi></mrow></msup></math></span> such that for every <span><math><mi>f</mi><mo>∈</mo><mi>Γ</mi></math></span> there is a set <span><math><mi>A</mi><mo>∈</mo><mi>A</mi></math></span> which is free for <em>f</em>, and <span><math><msub><mrow><mi>Δ</mi></mrow><mrow><mi>Γ</mi></mrow></msub></math></span> as the smallest size of a family <span><math><mi>F</mi><mo>⊆</mo><mi>Γ</mi></math></span> such that for every <span><math><mi>A</mi><mo>∈</mo><msup><mrow><mo>[</mo><mi>ω</mi><mo>]</mo></mrow><mrow><mi>ω</mi></mrow></msup></math></span> there is <span><math><mi>f</mi><mo>∈</mo><mi>F</mi></math></span> such that <em>A</em> is not free for <em>f</em>. We compare several versions of these cardinal invariants with some of the classical cardinal characteristics of the continuum. Using these notions, we partially answer some questions from <span><span>[20]</span></span> and <span><span>[2]</span></span>.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 8","pages":"Article 103607"},"PeriodicalIF":0.6,"publicationDate":"2025-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143922505","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}
Vera Fischer , Sy David Friedman , David Schrittesser , Asger Törnquist
{"title":"Good projective witnesses","authors":"Vera Fischer , Sy David Friedman , David Schrittesser , Asger Törnquist","doi":"10.1016/j.apal.2025.103606","DOIUrl":"10.1016/j.apal.2025.103606","url":null,"abstract":"<div><div>We develop a new forcing notion for adjoining self-coding cofinitary permutations and use it to show that consistently, the minimal cardinality <span><math><msub><mrow><mi>a</mi></mrow><mrow><mtext>g</mtext></mrow></msub></math></span> of a maximal cofinitary group (MCG) is strictly between <span><math><msub><mrow><mi>ℵ</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> and <span><math><mi>c</mi></math></span>, and there is a <span><math><msubsup><mrow><mi>Π</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>1</mn></mrow></msubsup></math></span>-definable MCG of this cardinality. Here <span><math><msubsup><mrow><mi>Π</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>1</mn></mrow></msubsup></math></span> is optimal, making this result a natural counterpart to the Borel MCG of Horowitz and Shelah. Our theorem has its analogue in the realm of maximal almost disjoint (MAD) families, extending a line of results regarding the definability properties of MAD families in models with large continuum.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 8","pages":"Article 103606"},"PeriodicalIF":0.6,"publicationDate":"2025-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143927371","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":"Weakly o-minimal types","authors":"Slavko Moconja , Predrag Tanović","doi":"10.1016/j.apal.2025.103605","DOIUrl":"10.1016/j.apal.2025.103605","url":null,"abstract":"<div><div>We introduce and study weak o-minimality in the context of complete types in an arbitrary first-order theory. A type <span><math><mi>p</mi><mo>∈</mo><mi>S</mi><mo>(</mo><mi>A</mi><mo>)</mo></math></span> is weakly o-minimal if for some relatively <em>A</em>-definable linear order, <, on <span><math><mi>p</mi><mo>(</mo><mi>C</mi><mo>)</mo></math></span> every relatively <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>C</mi></mrow></msub></math></span>-definable subset of <span><math><mi>p</mi><mo>(</mo><mi>C</mi><mo>)</mo></math></span> has finitely many convex components in <span><math><mo>(</mo><mi>p</mi><mo>(</mo><mi>C</mi><mo>)</mo><mo>,</mo><mo><</mo><mo>)</mo></math></span>. We establish many nice properties of weakly o-minimal types. For example, we prove that weakly o-minimal types are dp-minimal and share several properties of weight-one types in stable theories, and that a version of monotonicity theorem holds for relatively definable functions on the locus of a weakly o-minimal type.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 9","pages":"Article 103605"},"PeriodicalIF":0.6,"publicationDate":"2025-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143922216","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":"Unreachability of Γ2n+1,m","authors":"Derek Levinson","doi":"10.1016/j.apal.2025.103604","DOIUrl":"10.1016/j.apal.2025.103604","url":null,"abstract":"<div><div>We find bounds for the maximal length of a sequence of distinct <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mn>2n</mn><mo>+</mo><mn>1</mn><mo>,</mo><mi>m</mi></mrow></msub></math></span>-sets under <em>AD</em> and show there is no sequence of distinct <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mn>2n</mn><mo>+</mo><mn>1</mn></mrow></msub></math></span>-sets of length <span><math><msubsup><mrow><mi>δ</mi></mrow><mrow><mn>2n</mn><mo>+</mo><mn>3</mn></mrow><mrow><mn>1</mn></mrow></msubsup></math></span>. As a special case, there is no sequence of distinct <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>m</mi></mrow></msub></math></span>-sets of length <span><math><msub><mrow><mi>ℵ</mi></mrow><mrow><mi>m</mi><mo>+</mo><mn>2</mn></mrow></msub></math></span>. These are the optimal results for the pointclasses <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mn>2n</mn><mo>+</mo><mn>1</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>m</mi></mrow></msub></math></span>.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 9","pages":"Article 103604"},"PeriodicalIF":0.6,"publicationDate":"2025-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143895035","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":"Metric spaces in choiceless set theory","authors":"Eleftherios Tachtsis","doi":"10.1016/j.apal.2025.103603","DOIUrl":"10.1016/j.apal.2025.103603","url":null,"abstract":"<div><div>We <em>answer open questions</em> from Keremedis (2016) <span><span>[12]</span></span> and Keremedis and Tachtsis (2022) <span><span>[16]</span></span>, and <em>properly strengthen some results</em> from the above papers as well as from Keremedis et al. (2023) <span><span>[19]</span></span>. In particular, and among other results, we establish the following:<ul><li><span>1.</span><span><div>The Boolean Prime Ideal Theorem does not imply “For every sequentially compact metric space <span><math><mo>〈</mo><mi>X</mi><mo>,</mo><mi>d</mi><mo>〉</mo></math></span>, <span><math><mo>|</mo><mi>X</mi><mo>|</mo><mo>≤</mo><msup><mrow><mn>2</mn></mrow><mrow><msub><mrow><mi>ℵ</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></msup></math></span>” in <strong>ZF</strong> (Zermelo–Fraenkel set theory without the Axiom of Choice (<strong>AC</strong>)).</div></span></li><li><span>2.</span><span><div>“Every linearly ordered set can be well ordered” ∧ “The union of a well-orderable family of well-orderable sets is well orderable” ∧ “For every uncountable sequentially compact metric space <span><math><mo>〈</mo><mi>X</mi><mo>,</mo><mi>d</mi><mo>〉</mo></math></span>, <span><math><mo>|</mo><mi>X</mi><mo>|</mo><mo>=</mo><msup><mrow><mn>2</mn></mrow><mrow><msub><mrow><mi>ℵ</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></msup></math></span>” does not imply the Axiom of Countable Choice in <strong>ZFA</strong> (<strong>ZF</strong> with atoms).</div></span></li><li><span>3.</span><span><div>“For every uncountable compact metric space <span><math><mo>〈</mo><mi>X</mi><mo>,</mo><mi>d</mi><mo>〉</mo></math></span>, <span><math><mo>|</mo><mi>X</mi><mo>|</mo><mo>≥</mo><msup><mrow><mn>2</mn></mrow><mrow><msub><mrow><mi>ℵ</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></msup></math></span>” does not imply “For every uncountable sequentially compact metric space <span><math><mo>〈</mo><mi>X</mi><mo>,</mo><mi>d</mi><mo>〉</mo></math></span>, <span><math><mo>|</mo><mi>X</mi><mo>|</mo><mo>≥</mo><msup><mrow><mn>2</mn></mrow><mrow><msub><mrow><mi>ℵ</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></msup></math></span>” in <strong>ZFA</strong>”.</div></span></li><li><span>4.</span><span><div>“For every uncountable sequentially compact metric space <span><math><mo>〈</mo><mi>X</mi><mo>,</mo><mi>d</mi><mo>〉</mo></math></span> <span><math><mo>|</mo><mi>X</mi><mo>|</mo><mo>≥</mo><msup><mrow><mn>2</mn></mrow><mrow><msub><mrow><mi>ℵ</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></msup></math></span>” does not imply “For every uncountable compact metric space <span><math><mo>〈</mo><mi>X</mi><mo>,</mo><mi>d</mi><mo>〉</mo></math></span>, <span><math><mo>|</mo><mi>X</mi><mo>|</mo><mo>=</mo><msup><mrow><mn>2</mn></mrow><mrow><msub><mrow><mi>ℵ</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></msup></math></span>” in <strong>ZFA</strong>.</div></span></li><li><span>5.</span><span><div>“For every uncountable sequentially compact metric space <span><math><mo>〈</mo><mi>X</mi><mo>,</mo><mi>d</mi><mo>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 9","pages":"Article 103603"},"PeriodicalIF":0.6,"publicationDate":"2025-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143886674","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":"Completeness in local positive logic","authors":"Arturo Rodríguez Fanlo , Ori Segel","doi":"10.1016/j.apal.2025.103601","DOIUrl":"10.1016/j.apal.2025.103601","url":null,"abstract":"<div><div>We develop the basic model theory of local positive logic, a new logic that mixes positive logic (where negation is not allowed) and local logic (where models omit types of infinite distant pairs). We study several basic model theoretic notions such as compactness, positive closedness (existential closedness) and completeness (irreducibility).</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 7","pages":"Article 103601"},"PeriodicalIF":0.6,"publicationDate":"2025-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143820322","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}
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