Vera Fischer , L. Schembecker , David Schrittesser
{"title":"Tight cofinitary groups","authors":"Vera Fischer , L. Schembecker , David Schrittesser","doi":"10.1016/j.apal.2025.103570","DOIUrl":"10.1016/j.apal.2025.103570","url":null,"abstract":"<div><div>We introduce the notion of a tight cofinitary group, which captures forcing indestructibility of maximal cofinitary groups for a long list of partial orders, including Cohen, Sacks, Miller, Miller partition forcing and Shelah's poset for diagonalizing maximal ideals. Introducing a new robust coding technique, we establish the relative consistency of <span><math><msub><mrow><mi>a</mi></mrow><mrow><mi>g</mi></mrow></msub><mo>=</mo><mi>d</mi><mo><</mo><mi>c</mi><mo>=</mo><msub><mrow><mi>ℵ</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> alongside the existence of a <span><math><msubsup><mrow><mi>Δ</mi></mrow><mrow><mn>3</mn></mrow><mrow><mn>1</mn></mrow></msubsup></math></span>-well-order of the reals and a co-analytic witness for <span><math><msub><mrow><mi>a</mi></mrow><mrow><mi>g</mi></mrow></msub></math></span>.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 6","pages":"Article 103570"},"PeriodicalIF":0.6,"publicationDate":"2025-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143601168","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":"A finitary Kronecker's lemma and large deviations in the strong law of large numbers on Banach spaces","authors":"Morenikeji Neri","doi":"10.1016/j.apal.2025.103569","DOIUrl":"10.1016/j.apal.2025.103569","url":null,"abstract":"<div><div>We explore the computational content of Kronecker's lemma via the proof-theoretic perspective of proof mining and utilise the resulting finitary variant of this fundamental result to provide new rates for the Strong Law of Large Numbers for random variables taking values in type <em>p</em> Banach spaces, which in particular are very uniform in the sense that they do not depend on the distribution of the random variables. Furthermore, we provide computability-theoretic arguments to demonstrate the ineffectiveness of Kronecker's lemma and investigate the result from the perspective of Reverse Mathematics. In addition, we demonstrate how this ineffectiveness from Kronecker's lemma trickles down to the Strong Law of Large Numbers by providing a construction that shows that computable rates of convergence are not always possible. Lastly, we demonstrate how Kronecker's lemma falls under a class of deterministic formulas whose solution to their Dialectica interpretation satisfies a continuity property and how, for such formulas, one obtains an upgrade principle that allows one to lift computational interpretations of deterministic results to quantitative results for their probabilistic analogue. This result generalises the previous work of the author and Pischke.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 6","pages":"Article 103569"},"PeriodicalIF":0.6,"publicationDate":"2025-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143562528","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":"Universally Sacks-indestructible combinatorial families of reals","authors":"V. Fischer , L. Schembecker","doi":"10.1016/j.apal.2025.103566","DOIUrl":"10.1016/j.apal.2025.103566","url":null,"abstract":"<div><div>We introduce the notion of an arithmetical type of combinatorial family of reals, which serves to generalize different types of families such as mad families, maximal cofinitary groups, ultrafilter bases, splitting families and other similar types of families commonly studied in combinatorial set theory.</div><div>We then prove that every combinatorial family of reals of arithmetical type which is indestructible by the product of Sacks forcing <span><math><msup><mrow><mi>S</mi></mrow><mrow><msub><mrow><mi>ℵ</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></msup></math></span> is in fact universally Sacks-indestructible, i.e. it is indestructible by any countably supported iteration or product of Sacks-forcing of any length. Further, under <span><math><mi>CH</mi></math></span> we present a unified construction of universally Sacks-indestructible families for various arithmetical types of families. In particular we prove the existence of a universally Sacks-indestructible maximal cofinitary group under <span><math><mi>CH</mi></math></span>.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 6","pages":"Article 103566"},"PeriodicalIF":0.6,"publicationDate":"2025-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143562527","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":"Iterated reduced powers of collapsing algebras","authors":"Miloš S. Kurilić","doi":"10.1016/j.apal.2025.103567","DOIUrl":"10.1016/j.apal.2025.103567","url":null,"abstract":"<div><div><span><math><mrow><mi>rp</mi></mrow><mo>(</mo><mi>B</mi><mo>)</mo></math></span> denotes the reduced power <span><math><msup><mrow><mi>B</mi></mrow><mrow><mi>ω</mi></mrow></msup><mo>/</mo><mi>Φ</mi></math></span> of a Boolean algebra <span><math><mi>B</mi></math></span>, where Φ is the Fréchet filter on <em>ω</em>. We investigate iterated reduced powers (<span><math><msup><mrow><mi>rp</mi></mrow><mrow><mn>0</mn></mrow></msup><mo>(</mo><mi>B</mi><mo>)</mo><mo>=</mo><mi>B</mi></math></span> and <span><math><msup><mrow><mi>rp</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>(</mo><mi>B</mi><mo>)</mo><mo>=</mo><mrow><mi>rp</mi></mrow><mo>(</mo><msup><mrow><mi>rp</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>(</mo><mi>B</mi><mo>)</mo><mo>)</mo></math></span>) of collapsing algebras and our main intention is to classify the algebras <span><math><msup><mrow><mi>rp</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>(</mo><mrow><mi>Col</mi></mrow><mo>(</mo><mi>λ</mi><mo>,</mo><mi>κ</mi><mo>)</mo><mo>)</mo></math></span>, <span><math><mi>n</mi><mo>≥</mo><mn>1</mn></math></span>, up to isomorphism of their Boolean completions. In particular, assuming that SCH and <span><math><mi>h</mi><mo>=</mo><msub><mrow><mi>ω</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> hold, we show that for any cardinals <span><math><mi>λ</mi><mo>≥</mo><mi>ω</mi></math></span> and <span><math><mi>κ</mi><mo>≥</mo><mn>2</mn></math></span> such that <span><math><mi>κ</mi><mi>λ</mi><mo>></mo><mi>ω</mi></math></span> and <span><math><mrow><mi>cf</mi></mrow><mo>(</mo><mi>λ</mi><mo>)</mo><mo>≤</mo><mi>c</mi></math></span> we have <span><math><mrow><mi>ro</mi></mrow><mo>(</mo><msup><mrow><mi>rp</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>(</mo><mrow><mi>Col</mi></mrow><mo>(</mo><mi>λ</mi><mo>,</mo><mi>κ</mi><mo>)</mo><mo>)</mo><mo>)</mo><mo>≅</mo><mrow><mi>Col</mi></mrow><mo>(</mo><msub><mrow><mi>ω</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msup><mrow><mo>(</mo><msup><mrow><mi>κ</mi></mrow><mrow><mo><</mo><mi>λ</mi></mrow></msup><mo>)</mo></mrow><mrow><mi>ω</mi></mrow></msup><mo>)</mo></math></span>, for each <span><math><mi>n</mi><mo>≥</mo><mn>1</mn></math></span>; more precisely,<span><span><span><math><mrow><mi>ro</mi></mrow><mo>(</mo><msup><mrow><mi>rp</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>(</mo><mrow><mi>Col</mi></mrow><mo>(</mo><mi>λ</mi><mo>,</mo><mi>κ</mi><mo>)</mo><mo>)</mo><mo>)</mo><mo>≅</mo><mrow><mo>{</mo><mtable><mtr><mtd><mrow><mi>Col</mi></mrow><mo>(</mo><msub><mrow><mi>ω</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mi>c</mi><mo>)</mo><mo>,</mo><mspace></mspace></mtd><mtd><mtext> if </mtext><msup><mrow><mi>κ</mi></mrow><mrow><mo><</mo><mi>λ</mi></mrow></msup><mo>≤</mo><mi>c</mi><mo>;</mo></mtd></mtr><mtr><mtd><mrow><mi>Col</mi></mrow><mo>(</mo><msub><mrow><mi>ω</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msup><mrow><mi>κ</mi></mrow><mrow><mo><</mo><mi>λ</mi></mrow></msup><mo>)</mo><mo>,</mo><mspace></mspa","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 6","pages":"Article 103567"},"PeriodicalIF":0.6,"publicationDate":"2025-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143547846","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":"Automorphism groups of prime models, and invariant measures","authors":"Anand Pillay","doi":"10.1016/j.apal.2025.103568","DOIUrl":"10.1016/j.apal.2025.103568","url":null,"abstract":"<div><div>We adapt the notion from <span><span>[7]</span></span> and <span><span>[2]</span></span> of a (relatively) definable subset of <span><math><mi>A</mi><mi>u</mi><mi>t</mi><mo>(</mo><mi>M</mi><mo>)</mo></math></span> when <em>M</em> is a saturated structure, to the case <span><math><mi>A</mi><mi>u</mi><mi>t</mi><mo>(</mo><mi>M</mi><mo>/</mo><mi>A</mi><mo>)</mo></math></span> when <em>M</em> is atomic and strongly <em>ω</em>-homogeneous (over a set <em>A</em>). We discuss the existence and uniqueness of invariant measures on the Boolean algebra of definable subsets of <span><math><mi>A</mi><mi>u</mi><mi>t</mi><mo>(</mo><mi>M</mi><mo>/</mo><mi>A</mi><mo>)</mo></math></span>. For example when <em>T</em> is stable, we have existence and uniqueness.</div><div>We also discuss the compatibility of our definability notions with definable Galois cohomology from <span><span>[12]</span></span> and differential Galois theory.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 6","pages":"Article 103568"},"PeriodicalIF":0.6,"publicationDate":"2025-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143547845","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":"Borel sets without perfectly many overlapping translations, III","authors":"Andrzej Rosłanowski , Saharon Shelah","doi":"10.1016/j.apal.2025.103565","DOIUrl":"10.1016/j.apal.2025.103565","url":null,"abstract":"<div><div>We expand the results of Rosłanowski and Shelah <span><span>[11]</span></span>, <span><span>[10]</span></span> to all perfect Abelian Polish groups <span><math><mo>(</mo><mi>H</mi><mo>,</mo><mo>+</mo><mo>)</mo></math></span>. In particular, we show that if <span><math><mi>α</mi><mo><</mo><msub><mrow><mi>ω</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> and <span><math><mn>4</mn><mo>≤</mo><mi>k</mi><mo><</mo><mi>ω</mi></math></span>, then there is a ccc forcing notion adding a <span><math><msubsup><mrow><mi>Σ</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>0</mn></mrow></msubsup></math></span> set <span><math><mi>B</mi><mo>⊆</mo><mi>H</mi></math></span> which has <span><math><msub><mrow><mi>ℵ</mi></mrow><mrow><mi>α</mi></mrow></msub></math></span> many pairwise <em>k</em>–overlapping translations but not a perfect set of such translations. The technicalities of the forcing construction led us to investigations of the question when, in an Abelian group, <span><math><mi>X</mi><mo>−</mo><mi>X</mi><mo>⊆</mo><mi>Y</mi><mo>−</mo><mi>Y</mi></math></span> imply that a translation of <em>X</em> or −<em>X</em> is included in <em>Y</em>.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 6","pages":"Article 103565"},"PeriodicalIF":0.6,"publicationDate":"2025-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143547844","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":"Comparing notions of presentability in Polish spaces and Polish groups","authors":"Sapir Ben-Shahar , Heer Tern Koh","doi":"10.1016/j.apal.2025.103564","DOIUrl":"10.1016/j.apal.2025.103564","url":null,"abstract":"<div><div>A recent area of interest in computable topology compares different notions of effective presentability for topological spaces. In this paper, we show that up to isometry, there is a compact connected Polish space that has both left-c.e. and right-c.e. Polish presentations, but has no computable Polish presentation. We also construct a Polish group that has both left-c.e. and right-c.e. Polish group presentations, but lacks a computable Polish presentation, up to topological isomorphism.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 5","pages":"Article 103564"},"PeriodicalIF":0.6,"publicationDate":"2025-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143463576","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":"Local tabularity is decidable for bi-intermediate logics of trees and of co-trees","authors":"Miguel Martins, Tommaso Moraschini","doi":"10.1016/j.apal.2025.103563","DOIUrl":"10.1016/j.apal.2025.103563","url":null,"abstract":"<div><div>A bi-Heyting algebra validates the Gödel-Dummett axiom <span><math><mo>(</mo><mi>p</mi><mo>→</mo><mi>q</mi><mo>)</mo><mo>∨</mo><mo>(</mo><mi>q</mi><mo>→</mo><mi>p</mi><mo>)</mo></math></span> iff the poset of its prime filters is a disjoint union of co-trees (i.e., order duals of trees). Bi-Heyting algebras of this kind are called <em>bi-Gödel algebras</em> and form a variety that algebraizes the extension <span><math><mi>bi-GD</mi></math></span> of bi-intuitionistic logic axiomatized by the Gödel-Dummett axiom. In this paper we establish the decidability of the problem of determining if a finitely axiomatizable extension of <span><math><mi>bi-GD</mi></math></span> is locally tabular.</div><div>Notably, if <em>L</em> is an axiomatic extension of <span><math><mi>bi-GD</mi></math></span>, then <em>L</em> is locally tabular iff <em>L</em> is not contained in <span><math><mi>L</mi><mi>o</mi><mi>g</mi><mo>(</mo><mi>F</mi><mi>C</mi><mo>)</mo></math></span>, the logic of a particular family of finite co-trees, called the <em>finite combs</em>. We prove that <span><math><mi>L</mi><mi>o</mi><mi>g</mi><mo>(</mo><mi>F</mi><mi>C</mi><mo>)</mo></math></span> is finitely axiomatizable. Since this logic also has the finite model property, it is therefore decidable. Thus, the above characterization of local tabularity ensures the decidability of the aforementioned problem.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 5","pages":"Article 103563"},"PeriodicalIF":0.6,"publicationDate":"2025-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143487726","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}
Sergio Celani , Rafał Gruszczyński , Paula Menchón
{"title":"Conditional algebras","authors":"Sergio Celani , Rafał Gruszczyński , Paula Menchón","doi":"10.1016/j.apal.2025.103556","DOIUrl":"10.1016/j.apal.2025.103556","url":null,"abstract":"<div><div>Drawing on the classic paper by Chellas <span><span>[8]</span></span>, we propose a general algebraic framework for studying a binary operation of <em>conditional</em> that models universal features of the “if …, then …” connective as strictly related to the unary modal necessity operator. To this end, we introduce a variety of <em>conditional algebras</em>, and we develop its duality and canonical extensions theory.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 5","pages":"Article 103556"},"PeriodicalIF":0.6,"publicationDate":"2025-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143376529","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":"Proof-theoretic methods in quantifier-free definability","authors":"Zoltan A. Kocsis","doi":"10.1016/j.apal.2025.103555","DOIUrl":"10.1016/j.apal.2025.103555","url":null,"abstract":"<div><div>We introduce a proof-theoretic approach to showing nondefinability of second-order intuitionistic connectives by quantifier-free schemata. We apply the method to prove that Taranovsky's “realizability disjunction” connective does not admit a quantifier-free definition, and use it to obtain new results and more nuanced information about the nondefinability of Kreisel's and Połacik's unary connectives. The finitary and combinatorial nature of our method makes it resilient to changes in metatheory, and suitable for settings with axioms that are explicitly incompatible with classical logic. Furthermore, the problem-specific subproofs arising from this approach can be readily transcribed into univalent type theory and verified using the Agda proof assistant.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 4","pages":"Article 103555"},"PeriodicalIF":0.6,"publicationDate":"2025-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143162322","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}
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