Pub Date : 2025-12-05DOI: 10.1016/j.apal.2025.103704
Thomas Koberda, Yash Lodha
We use model theoretic forcing to prove that a generic countable torsion-free group does not admit any non-trivial locally moving action on a Hausdorff topological space, and yet admits a rich Rubin poset.
{"title":"Generic torsion-free groups and Rubin actions","authors":"Thomas Koberda, Yash Lodha","doi":"10.1016/j.apal.2025.103704","DOIUrl":"10.1016/j.apal.2025.103704","url":null,"abstract":"<div><div>We use model theoretic forcing to prove that a generic countable torsion-free group does not admit any non-trivial locally moving action on a Hausdorff topological space, and yet admits a rich Rubin poset.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"177 5","pages":"Article 103704"},"PeriodicalIF":0.6,"publicationDate":"2025-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145738714","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}
Pub Date : 2025-12-02DOI: 10.1016/j.apal.2025.103695
Evelina Daniyarova , Alexei Myasnikov
We prove that metabelian Baumslag – Solitar group , , is (strongly) regularly bi-interpretable with the ring of integers , and describe in algebraic terms all groups that are elementarily equivalent to .
{"title":"Groups elementarily equivalent to metabelian Baumslag – Solitar groups and regular bi-interpretability","authors":"Evelina Daniyarova , Alexei Myasnikov","doi":"10.1016/j.apal.2025.103695","DOIUrl":"10.1016/j.apal.2025.103695","url":null,"abstract":"<div><div>We prove that metabelian Baumslag<!--> <!-->–<!--> <!-->Solitar group <span><math><mrow><mi>BS</mi></mrow><mo>(</mo><mn>1</mn><mo>,</mo><mi>k</mi><mo>)</mo></math></span>, <span><math><mi>k</mi><mo>></mo><mn>1</mn></math></span>, is (strongly) regularly bi-interpretable with the ring of integers <span><math><mi>Z</mi></math></span>, and describe in algebraic terms all groups that are elementarily equivalent to <span><math><mrow><mi>BS</mi></mrow><mo>(</mo><mn>1</mn><mo>,</mo><mi>k</mi><mo>)</mo></math></span>.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"177 5","pages":"Article 103695"},"PeriodicalIF":0.6,"publicationDate":"2025-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145658449","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}
Pub Date : 2025-11-27DOI: 10.1016/j.apal.2025.103694
Jonathan Schilhan
We show that it is consistent relative to , that there is no well-ordering of while a wide class of special sets of reals such as Hamel bases, transcendence bases, Vitali sets or Bernstein sets exists. To be more precise, we can assume that every projective hypergraph on has a maximal independent set, among a few other things. For example, we get transversals for all projective equivalence relations. Moreover, this is possible while either holds, or countable choice for reals fails. Assuming the consistency of an inaccessible cardinal, “projective” can even be replaced with “” and we can add that any instance of in has a choice function. This vastly strengthens the consistency results obtained in [6], [11] or [15].
{"title":"Maximal sets without choice","authors":"Jonathan Schilhan","doi":"10.1016/j.apal.2025.103694","DOIUrl":"10.1016/j.apal.2025.103694","url":null,"abstract":"<div><div>We show that it is consistent relative to <span><math><mi>ZF</mi></math></span>, that there is no well-ordering of <span><math><mi>R</mi></math></span> while a wide class of special sets of reals such as Hamel bases, transcendence bases, Vitali sets or Bernstein sets exists. To be more precise, we can assume that every projective hypergraph on <span><math><mi>R</mi></math></span> has a maximal independent set, among a few other things. For example, we get transversals for all projective equivalence relations. Moreover, this is possible while either <span><math><msub><mrow><mi>DC</mi></mrow><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub></math></span> holds, or countable choice for reals fails. Assuming the consistency of an inaccessible cardinal, “projective” can even be replaced with “<span><math><mi>L</mi><mo>(</mo><mi>R</mi><mo>)</mo></math></span>” and we can add that any instance of <span><math><mi>AC</mi></math></span> in <span><math><mi>L</mi><mo>(</mo><mi>R</mi><mo>)</mo></math></span> has a choice function. This vastly strengthens the consistency results obtained in <span><span>[6]</span></span>, <span><span>[11]</span></span> or <span><span>[15]</span></span>.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"177 4","pages":"Article 103694"},"PeriodicalIF":0.6,"publicationDate":"2025-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145694156","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}
Pub Date : 2025-11-27DOI: 10.1016/j.apal.2025.103693
Marie Duží , Bjørn Jespersen
We address and solve some technical problems arising in Transparent Intensional Logic, which identifies hyperintensions with algorithmically structured procedures. The problems concern variable binding and substitution. A pair of procedures, called Trivialization and Double Execution, work together well in most cases. But there are limiting cases where they fail to. Trivialization, if applied to another procedure, makes it feasible to operate on this procedure itself rather than on the product it is typed and structured to yield. Any occurrences of variables within the procedure become Trivialization-bound. Trivialization-binding is stronger than λ-binding. Double Execution, when applied to another procedure, is typed and structured to, first, obtain the product of this procedure and, second, obtain the product of this product on condition that the latter is itself a procedure. Double Execution can turn some Trivialization-bound occurrences of variables into free occurrences, and Double Execution may also produce fresh variables. Such cases undermine our definition of substitution, thus jeopardizing the validity of the rules of λ-conversion. The restrictions required are obtained by revising fundamental definitions. The problem we address and solve extends beyond Transparent Intensional Logic. Any logic or programming language (especially if based on the λ-calculus) furnished with a high degree of expressive power, in which procedures can occur as operands, is liable to confront similar problems.
{"title":"Twin procedures, two kinds of variable binding, and two kinds of computation","authors":"Marie Duží , Bjørn Jespersen","doi":"10.1016/j.apal.2025.103693","DOIUrl":"10.1016/j.apal.2025.103693","url":null,"abstract":"<div><div>We address and solve some technical problems arising in Transparent Intensional Logic, which identifies hyperintensions with algorithmically structured procedures. The problems concern variable binding and substitution. A pair of procedures, called Trivialization and Double Execution, work together well in most cases. But there are limiting cases where they fail to. Trivialization, if applied to another procedure, makes it feasible to operate on this procedure itself rather than on the product it is typed and structured to yield. Any occurrences of variables within the procedure become Trivialization-bound. Trivialization-binding is stronger than <em>λ</em>-binding. Double Execution, when applied to another procedure, is typed and structured to, first, obtain the product of this procedure and, second, obtain the product of this product on condition that the latter is itself a procedure. Double Execution can turn some Trivialization-bound occurrences of variables into free occurrences, and Double Execution may also produce fresh variables. Such cases undermine our definition of substitution, thus jeopardizing the validity of the rules of <em>λ</em>-conversion. The restrictions required are obtained by revising fundamental definitions. The problem we address and solve extends beyond Transparent Intensional Logic. Any logic or programming language (especially if based on the <em>λ</em>-calculus) furnished with a high degree of expressive power, in which procedures can occur as operands, is liable to confront similar problems.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"177 4","pages":"Article 103693"},"PeriodicalIF":0.6,"publicationDate":"2025-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145797266","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}
Pub Date : 2025-11-19DOI: 10.1016/j.apal.2025.103683
Mark Saving
We formulate a definition of the existence property that works with “structural” set theories, in the mode of ETCS (the elementary theory of the category of sets). We show that a range of structural set theories, when formulated using constructive logic, satisfy the disjunction, numerical existence, and existence properties; in particular, intuitionist ETCS, formulated with separation and Shulman's Replacement of Contexts axiom, satisfies these properties. As a consequence of this, we show that, working constructively, Replacement of Contexts is strictly weaker than collection.
{"title":"On the existence and disjunction properties in structural set theory","authors":"Mark Saving","doi":"10.1016/j.apal.2025.103683","DOIUrl":"10.1016/j.apal.2025.103683","url":null,"abstract":"<div><div>We formulate a definition of the existence property that works with “structural” set theories, in the mode of ETCS (the elementary theory of the category of sets). We show that a range of structural set theories, when formulated using constructive logic, satisfy the disjunction, numerical existence, and existence properties; in particular, intuitionist ETCS, formulated with separation and Shulman's Replacement of Contexts axiom, satisfies these properties. As a consequence of this, we show that, working constructively, Replacement of Contexts is strictly weaker than collection.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"177 4","pages":"Article 103683"},"PeriodicalIF":0.6,"publicationDate":"2025-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145625142","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}
Pub Date : 2025-11-19DOI: 10.1016/j.apal.2025.103682
Françoise Point , Nathalie Regnault
We axiomatize a class of existentially closed differential expansions of exponential topological fields where the derivation is an E-derivation. We apply our results to differential expansions of, on the one hand the field of real numbers endowed with , the classical exponential function defined by its power series expansion, and on the other hand the field of p-adic numbers endowed with the function defined on the subring of p-adic integers where p is a prime number strictly bigger than 2 (or with when ).
{"title":"Exponential topological fields with a generic derivation","authors":"Françoise Point , Nathalie Regnault","doi":"10.1016/j.apal.2025.103682","DOIUrl":"10.1016/j.apal.2025.103682","url":null,"abstract":"<div><div>We axiomatize a class of existentially closed differential expansions of exponential topological fields where the derivation is an <em>E</em>-derivation. We apply our results to differential expansions of, on the one hand the field of real numbers endowed with <span><math><mi>e</mi><mi>x</mi><mi>p</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span>, the classical exponential function defined by its power series expansion, and on the other hand the field of p-adic numbers endowed with the function <span><math><mi>e</mi><mi>x</mi><mi>p</mi><mo>(</mo><mi>p</mi><mi>x</mi><mo>)</mo></math></span> defined on the subring of <em>p</em>-adic integers where <em>p</em> is a prime number strictly bigger than 2 (or with <span><math><mi>e</mi><mi>x</mi><mi>p</mi><mo>(</mo><mn>4</mn><mi>x</mi><mo>)</mo></math></span> when <span><math><mi>p</mi><mo>=</mo><mn>2</mn></math></span>).</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"177 4","pages":"Article 103682"},"PeriodicalIF":0.6,"publicationDate":"2025-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145584227","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}
Pub Date : 2025-11-11DOI: 10.1016/j.apal.2025.103679
Jorge Antonio Cruz Chapital, Osvaldo Guzmán González, Stevo Todorčević
A structural analysis of construction schemes is developed. That analysis is used to give simple and new constructions of combinatorial objects which have been of interest to set theorists and topologists. We then continue the study of capturing axioms associated to construction schemes. From them, we deduce the existence of several uncountable structures which are known to be independent from the usual axioms of Set Theory. Lastly, we prove that the capturing axiom is implied by Jensen's ◇ principle.
{"title":"Construction schemes: Transferring structures from ω to ω1","authors":"Jorge Antonio Cruz Chapital, Osvaldo Guzmán González, Stevo Todorčević","doi":"10.1016/j.apal.2025.103679","DOIUrl":"10.1016/j.apal.2025.103679","url":null,"abstract":"<div><div>A structural analysis of construction schemes is developed. That analysis is used to give simple and new constructions of combinatorial objects which have been of interest to set theorists and topologists. We then continue the study of capturing axioms associated to construction schemes. From them, we deduce the existence of several uncountable structures which are known to be independent from the usual axioms of Set Theory. Lastly, we prove that the capturing axiom <span><math><mi>F</mi><mi>C</mi><mi>A</mi><mo>(</mo><mi>p</mi><mi>a</mi><mi>r</mi><mi>t</mi><mo>)</mo></math></span> is implied by Jensen's ◇ principle.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"177 4","pages":"Article 103679"},"PeriodicalIF":0.6,"publicationDate":"2025-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145584226","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}
Pub Date : 2025-11-10DOI: 10.1016/j.apal.2025.103678
Yiqun Liu , Yong Liu , Cheng Peng
A Turing degree is d.c.e. if it contains a set that is the difference of two c.e. sets. A d.c.e. degree d is isolated by a c.e. degree if all c.e. degrees that are below d are also below a; d is isolated from above by a c.e. degree if all c.e. degrees that are above d are also above a. In this paper, we study the inductive strength of both isolated and upper isolated d.c.e. degrees from the point of view of reverse recursion theory. We show that (1) There is an isolated proper d.c.e. degree below ; (2) There is an upper isolated proper d.c.e. degree below .
{"title":"Isolated d.c.e. degrees and Σ1 induction","authors":"Yiqun Liu , Yong Liu , Cheng Peng","doi":"10.1016/j.apal.2025.103678","DOIUrl":"10.1016/j.apal.2025.103678","url":null,"abstract":"<div><div>A Turing degree is d.c.e. if it contains a set that is the difference of two c.e. sets. A d.c.e. degree <strong><em>d</em></strong> is <em>isolated</em> by a c.e. degree <span><math><mi>a</mi><mo><</mo><mi>d</mi></math></span> if all c.e. degrees that are below <strong><em>d</em></strong> are also below <strong><em>a</em></strong>; <strong><em>d</em></strong> is <em>isolated from above</em> by a c.e. degree <span><math><mi>a</mi><mo>></mo><mi>d</mi></math></span> if all c.e. degrees that are above <strong><em>d</em></strong> are also above <strong><em>a</em></strong>. In this paper, we study the inductive strength of both isolated and upper isolated d.c.e. degrees from the point of view of reverse recursion theory. We show that (1) <span><math><msup><mrow><mi>P</mi></mrow><mrow><mo>−</mo></mrow></msup><mo>+</mo><mi>B</mi><msub><mrow><mi>Σ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><mtext>Exp</mtext><mo>⊢</mo><mi>I</mi><msub><mrow><mi>Σ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>↔</mo></math></span> There is an isolated proper d.c.e. degree below <span><math><msup><mrow><mn>0</mn></mrow><mrow><mo>′</mo></mrow></msup></math></span>; (2) <span><math><msup><mrow><mi>P</mi></mrow><mrow><mo>−</mo></mrow></msup><mo>+</mo><mi>B</mi><msub><mrow><mi>Σ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><mtext>Exp</mtext><mo>⊢</mo><mi>I</mi><msub><mrow><mi>Σ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>↔</mo></math></span> There is an upper isolated proper d.c.e. degree below <span><math><msup><mrow><mn>0</mn></mrow><mrow><mo>′</mo></mrow></msup></math></span>.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"177 3","pages":"Article 103678"},"PeriodicalIF":0.6,"publicationDate":"2025-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145528622","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}
Pub Date : 2025-11-10DOI: 10.1016/j.apal.2025.103680
Witold Marciszewski, Roman Pol, Piotr Zakrzewski
The Hurewicz property is a classical generalization of σ-compactness. Sierpiński sets (whose existence follows from CH) are standard examples of non-σ-compact Hurewicz spaces. We show, solving a problem stated by Szewczak and Tsaban [19], that for each Sierpiński set S of cardinality at least there is a Hurewicz space H with not Hurewicz.
Some other questions in the literature concerning this topic are also answered.
{"title":"On Sierpiński sets, Hurewicz spaces and Hilgers functions","authors":"Witold Marciszewski, Roman Pol, Piotr Zakrzewski","doi":"10.1016/j.apal.2025.103680","DOIUrl":"10.1016/j.apal.2025.103680","url":null,"abstract":"<div><div>The Hurewicz property is a classical generalization of <em>σ</em>-compactness. Sierpiński sets (whose existence follows from CH) are standard examples of non-<em>σ</em>-compact Hurewicz spaces. We show, solving a problem stated by Szewczak and Tsaban <span><span>[19]</span></span>, that for each Sierpiński set <em>S</em> of cardinality at least <span><math><mi>b</mi></math></span> there is a Hurewicz space <em>H</em> with <span><math><mi>S</mi><mo>×</mo><mi>H</mi></math></span> not Hurewicz.</div><div>Some other questions in the literature concerning this topic are also answered.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"177 3","pages":"Article 103680"},"PeriodicalIF":0.6,"publicationDate":"2025-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145528621","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}
Pub Date : 2025-10-22DOI: 10.1016/j.apal.2025.103677
Adam Přenosil
Unlike in classical modal logic, in non-classical modal logics the box and diamond operators frequently fail to be interdefinable. Instead, these logics impose some compatibility conditions between the two operators which ensure that in terms of Kripke semantics they arise from the same accessibility relation. This is the case in the intuitionistic modal logic of Fischer Servi as well as the positive modal logic of Dunn. In these logics, however, such compatibility conditions also impose further restrictions on the accessibility relation. In this paper, we identify the basic compatibility conditions which ensure that modal operators arise from a single accessibility relation without imposing any restrictions on the relation. As in the distributive logic of Gehrke, Nagahashi, and Venema, we allow for negative box and diamond operators here in addition to the usual positive ones. Intuitionistic modal logic and positive modal logic, or more precisely the corresponding classes of algebras, are then obtained in a modular way by adding certain canonical axioms which we call locality conditions on top of these basic compatibility conditions.
{"title":"Compatibility between modal operators in distributive modal logic","authors":"Adam Přenosil","doi":"10.1016/j.apal.2025.103677","DOIUrl":"10.1016/j.apal.2025.103677","url":null,"abstract":"<div><div>Unlike in classical modal logic, in non-classical modal logics the box and diamond operators frequently fail to be interdefinable. Instead, these logics impose some compatibility conditions between the two operators which ensure that in terms of Kripke semantics they arise from the same accessibility relation. This is the case in the intuitionistic modal logic of Fischer Servi as well as the positive modal logic of Dunn. In these logics, however, such compatibility conditions also impose further restrictions on the accessibility relation. In this paper, we identify the basic compatibility conditions which ensure that modal operators arise from a single accessibility relation without imposing any restrictions on the relation. As in the distributive logic of Gehrke, Nagahashi, and Venema, we allow for negative box and diamond operators here in addition to the usual positive ones. Intuitionistic modal logic and positive modal logic, or more precisely the corresponding classes of algebras, are then obtained in a modular way by adding certain canonical axioms which we call locality conditions on top of these basic compatibility conditions.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"177 2","pages":"Article 103677"},"PeriodicalIF":0.6,"publicationDate":"2025-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145364462","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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