Pub Date : 2026-01-08DOI: 10.1016/j.apal.2026.103722
Lorenzo Notaro
We study the join-semilattice of constructibility real degrees in the side-by-side Sacks model, i.e., the model of set theory obtained by forcing with a countable-support product of infinitely many Sacks forcings over the constructible universe. In particular, we prove that in the side-by-side Sacks model the join-semilattice of constructibility real degrees is rigid, i.e., it does not have non-trivial automorphisms.
{"title":"Constructibility real degrees in the side-by-side Sacks model","authors":"Lorenzo Notaro","doi":"10.1016/j.apal.2026.103722","DOIUrl":"10.1016/j.apal.2026.103722","url":null,"abstract":"<div><div>We study the join-semilattice of constructibility real degrees in the side-by-side Sacks model, i.e., the model of set theory obtained by forcing with a countable-support product of infinitely many Sacks forcings over the constructible universe. In particular, we prove that in the side-by-side Sacks model the join-semilattice of constructibility real degrees is rigid, i.e., it does not have non-trivial automorphisms.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"177 5","pages":"Article 103722"},"PeriodicalIF":0.6,"publicationDate":"2026-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145976653","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 : 2026-01-08DOI: 10.1016/j.apal.2026.103723
Yifan Hu , Ruihuan Mao , Guozhen Shen
A set A is dually Dedekind finite if every surjection from A onto A is injective; otherwise, A is dually Dedekind infinite. An amorphous set is an infinite set that cannot be partitioned into two infinite subsets. A strictly amorphous set is an amorphous set in which every partition has only finitely many non-singleton blocks. It is proved consistent with (i.e., Zermelo–Fraenkel set theory without the axiom of choice) that there exists an amorphous set A whose power set is dually Dedekind infinite, which gives a negative solution to a question proposed by Truss (1974) [13]. Nevertheless, we prove in that, for all strictly amorphous sets A and all natural numbers n, is dually Dedekind finite, which generalizes a result of Goldstern.
{"title":"Amorphous sets and dual Dedekind finiteness","authors":"Yifan Hu , Ruihuan Mao , Guozhen Shen","doi":"10.1016/j.apal.2026.103723","DOIUrl":"10.1016/j.apal.2026.103723","url":null,"abstract":"<div><div>A set <em>A</em> is dually Dedekind finite if every surjection from <em>A</em> onto <em>A</em> is injective; otherwise, <em>A</em> is dually Dedekind infinite. An amorphous set is an infinite set that cannot be partitioned into two infinite subsets. A strictly amorphous set is an amorphous set in which every partition has only finitely many non-singleton blocks. It is proved consistent with <span><math><mi>ZF</mi></math></span> (i.e., Zermelo–Fraenkel set theory without the axiom of choice) that there exists an amorphous set <em>A</em> whose power set <span><math><mi>P</mi><mo>(</mo><mi>A</mi><mo>)</mo></math></span> is dually Dedekind infinite, which gives a negative solution to a question proposed by Truss (1974) <span><span>[13]</span></span>. Nevertheless, we prove in <span><math><mi>ZF</mi></math></span> that, for all strictly amorphous sets <em>A</em> and all natural numbers <em>n</em>, <span><math><mi>P</mi><msup><mrow><mo>(</mo><mi>A</mi><mo>)</mo></mrow><mrow><mi>n</mi></mrow></msup></math></span> is dually Dedekind finite, which generalizes a result of Goldstern.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"177 6","pages":"Article 103723"},"PeriodicalIF":0.6,"publicationDate":"2026-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145950127","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 : 2026-01-07DOI: 10.1016/j.apal.2025.103717
Alan Dow , Raúl Figueroa-Sierra , Osvaldo Guzmán , Michael Hrušák
We prove that there is an ideal on ω that is not Katětov below nwd and does not have restrictions above . We also prove that in the Laver model every tall P-ideal is Katětov-Blass above and that it is consistent that every Q+ ideal is meager.
{"title":"The category dichotomy for ideals","authors":"Alan Dow , Raúl Figueroa-Sierra , Osvaldo Guzmán , Michael Hrušák","doi":"10.1016/j.apal.2025.103717","DOIUrl":"10.1016/j.apal.2025.103717","url":null,"abstract":"<div><div>We prove that there is an ideal on <em>ω</em> that is not Katětov below <span>nwd</span> and does not have restrictions above <span><math><mi>ED</mi></math></span>. We also prove that in the Laver model every tall <span>P</span>-ideal is Katětov-Blass above <span><math><msub><mrow><mi>ED</mi></mrow><mrow><mtext>fin</mtext></mrow></msub></math></span> and that it is consistent that every <span>Q</span><sup>+</sup> ideal is meager.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"177 5","pages":"Article 103717"},"PeriodicalIF":0.6,"publicationDate":"2026-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145976652","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 : 2026-01-07DOI: 10.1016/j.apal.2026.103721
Osvaldo Guzmán , Francisco Santiago Nieto-de la Rosa
We analyze conditions to preserve with forcing multiple gaps, define conditions to ensure their existence, and study the minimal size of specific type of gaps.
分析了强制多间隙的保存条件，定义了保证多间隙存在的条件，研究了特定类型间隙的最小尺寸。
{"title":"Preserving and constructing multiple gaps","authors":"Osvaldo Guzmán , Francisco Santiago Nieto-de la Rosa","doi":"10.1016/j.apal.2026.103721","DOIUrl":"10.1016/j.apal.2026.103721","url":null,"abstract":"<div><div>We analyze conditions to preserve with forcing multiple gaps, define conditions to ensure their existence, and study the minimal size of specific type of gaps.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"177 5","pages":"Article 103721"},"PeriodicalIF":0.6,"publicationDate":"2026-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145976651","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 : 2026-01-02DOI: 10.1016/j.apal.2025.103707
Yutong Duan, Joel Nagloo
Let be distinct non-constant and non-degenerate solutions of the classical Lotka-Volterra system where . We show that if d and b are linearly independent over , then the solutions are algebraically independent over , that is . As a main part of the proof, we show that the set defined by the system in universal differential fields, with d and b linearly independent over , is strongly minimal and geometrically trivial. Our techniques also allows us to obtain partial results for some of the more general 2d-Lotka-Volterra system.
{"title":"Algebraic independence of the solutions of the classical Lotka-Volterra system","authors":"Yutong Duan, Joel Nagloo","doi":"10.1016/j.apal.2025.103707","DOIUrl":"10.1016/j.apal.2025.103707","url":null,"abstract":"<div><div>Let <span><math><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>,</mo><mo>…</mo><mo>,</mo><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></math></span> be distinct non-constant and non-degenerate solutions of the classical Lotka-Volterra system<span><span><span><math><msup><mrow><mi>x</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>=</mo><mi>a</mi><mi>x</mi><mi>y</mi><mo>+</mo><mi>b</mi><mi>x</mi><mspace></mspace><msup><mrow><mi>y</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>=</mo><mi>c</mi><mi>x</mi><mi>y</mi><mo>+</mo><mi>d</mi><mi>y</mi></math></span></span></span> where <span><math><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo>∈</mo><mi>C</mi><mo>∖</mo><mo>{</mo><mn>0</mn><mo>}</mo></math></span>. We show that if <em>d</em> and <em>b</em> are linearly independent over <span><math><mi>Q</mi></math></span>, then the solutions are algebraically independent over <span><math><mi>C</mi></math></span>, that is <span><math><mi>t</mi><mi>r</mi><mo>.</mo><mi>d</mi><mi>e</mi><msub><mrow><mi>g</mi></mrow><mrow><mi>C</mi></mrow></msub><mi>C</mi><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo><mo>=</mo><mn>2</mn><mi>n</mi></math></span>. As a main part of the proof, we show that the set defined by the system in universal differential fields, with <em>d</em> and <em>b</em> linearly independent over <span><math><mi>Q</mi></math></span>, is strongly minimal and geometrically trivial. Our techniques also allows us to obtain partial results for some of the more general 2<em>d</em>-Lotka-Volterra system.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"177 5","pages":"Article 103707"},"PeriodicalIF":0.6,"publicationDate":"2026-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145926009","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 : 2025-12-22DOI: 10.1016/j.apal.2025.103706
Max Illmer, Tim Netzer
We give a sufficient condition for a model theoretic structure B to ‘inherit’ quantifier elimination from another structure A. This yields an alternative proof of one of the main results from [8], namely quantifier elimination for certain matrix rings. The original proof uses model theory, and while it is very elegant and insightful, the proof we propose is much shorter and provides a constructive algorithm.
{"title":"Constructive quantifier elimination with a focus on matrix rings","authors":"Max Illmer, Tim Netzer","doi":"10.1016/j.apal.2025.103706","DOIUrl":"10.1016/j.apal.2025.103706","url":null,"abstract":"<div><div>We give a sufficient condition for a model theoretic structure <em>B</em> to ‘inherit’ quantifier elimination from another structure <em>A</em>. This yields an alternative proof of one of the main results from <span><span>[8]</span></span>, namely quantifier elimination for certain matrix rings. The original proof uses model theory, and while it is very elegant and insightful, the proof we propose is much shorter and provides a constructive algorithm.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"177 5","pages":"Article 103706"},"PeriodicalIF":0.6,"publicationDate":"2025-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145884440","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 : 2025-12-16DOI: 10.1016/j.apal.2025.103705
Yayi Fu
Using a result in [6] and the proof of [2, Theorem 1.1], we show that there is such that for any finite graph G with VC-dimension ≤2, G has a clique or an anti-clique of size . We also show that the Erdős-Hajnal property for graphs with VC-dimension 1 can be proved using the δ-dimension technique in [4], and we show that when E is a definable symmetric binary relation, [4, Theorem 1.3] can be proved without using Shelah's 2-rank.
{"title":"A note on Erdős-Hajnal property for graphs with VC dimension ≤2","authors":"Yayi Fu","doi":"10.1016/j.apal.2025.103705","DOIUrl":"10.1016/j.apal.2025.103705","url":null,"abstract":"<div><div>Using a result in <span><span>[6]</span></span> and the proof of <span><span>[2, Theorem 1.1]</span></span>, we show that there is <span><math><mi>γ</mi><mo>></mo><mn>0</mn></math></span> such that for any finite graph <em>G</em> with VC-dimension ≤2, <em>G</em> has a clique or an anti-clique of size <span><math><mo>≥</mo><mo>|</mo><mi>G</mi><msup><mrow><mo>|</mo></mrow><mrow><mi>γ</mi></mrow></msup></math></span>. We also show that the Erdős-Hajnal property for graphs with VC-dimension 1 can be proved using the <strong><em>δ</em></strong>-dimension technique in <span><span>[4]</span></span>, and we show that when <em>E</em> is a definable symmetric binary relation, <span><span>[4, Theorem 1.3]</span></span> can be proved without using Shelah's 2-rank.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"177 5","pages":"Article 103705"},"PeriodicalIF":0.6,"publicationDate":"2025-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145790897","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 : 2025-12-05DOI: 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}
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