Annals of Pure and Applied Logic最新文献

{"title":"Generic multiplicative endomorphism of a field","authors":"Christian d'Elbée","doi":"10.1016/j.apal.2025.103554","DOIUrl":"10.1016/j.apal.2025.103554","url":null,"abstract":"<div><div>We introduce the model-companion of the theory of fields expanded by a unary function for a multiplicative endomorphism, which we call ACFH. Among others, we prove that this theory is NSOP₁ and not simple, that the kernel of the map is a generic pseudo-finite abelian group. We also prove that if forking satisfies existence, then ACFH has elimination of imaginaries.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 4","pages":"Article 103554"},"PeriodicalIF":0.6,"publicationDate":"2025-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143162054","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":"Π2-rule systems and inductive classes of Gödel algebras","authors":"Rodrigo Nicolau Almeida","doi":"10.1016/j.apal.2025.103552","DOIUrl":"10.1016/j.apal.2025.103552","url":null,"abstract":"<div><div>In this paper we present a general theory of <span><math><msub><mrow><mi>Π</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-rules for systems of intuitionistic and modal logic. We introduce the notions of <span><math><msub><mrow><mi>Π</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-rule system and of an inductive class, and provide model-theoretic and algebraic completeness theorems, which serve as our basic tools. As an illustration of the general theory, we analyse the structure of inductive classes of Gödel algebras, from a structure theoretic and logical point of view. We show that unlike other well-studied settings (such as logics, or single-conclusion rule systems), there are continuum many <span><math><msub><mrow><mi>Π</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-rule systems extending <span><math><mrow><mi>LC</mi></mrow><mo>=</mo><mrow><mi>IPC</mi></mrow><mo>+</mo><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>, and show how our methods allow easy proofs of the admissibility of the well-known Takeuti-Titani rule. Our final results concern general questions admissibility in <span><math><mi>LC</mi></math></span>: (1) we present a full classification of those inductive classes which are inductively complete, i.e., where all <span><math><msub><mrow><mi>Π</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-rules which are admissible are derivable, and (2) show that the problem of admissibility of <span><math><msub><mrow><mi>Π</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-rules over <span><math><mi>LC</mi></math></span> is decidable.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 4","pages":"Article 103552"},"PeriodicalIF":0.6,"publicationDate":"2025-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143162347","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":"Modal logics over lattices","authors":"Xiaoyang Wang ,&nbsp;Yanjing Wang","doi":"10.1016/j.apal.2025.103553","DOIUrl":"10.1016/j.apal.2025.103553","url":null,"abstract":"<div><div>Lattice theory has various close connections with modal logic. However, one less explored direction is to view lattices as relational structures based on partial orders, and study the modal logics over them. In this paper, following the earlier steps of Burgess and van Benthem in the 1980s, we use the modal languages of tense logic and polyadic modal logic to talk about lattices via standard Kripke semantics. We first obtain a series of complete axiomatizations of tense logics over lattices, (un)bounded lattices over partial orders or strict orders. In particular, we solve an axiomatization problem left open by Burgess (1984) <span><span>[8]</span></span>. The second half of the paper gives a series of complete axiomatizations of polyadic modal logic with nominals over lattices, distributive lattices, and modular lattices, where the binary modalities of <em>infimum</em> and <em>supremum</em> can reveal more structures behind various lattices.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 4","pages":"Article 103553"},"PeriodicalIF":0.6,"publicationDate":"2025-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143162323","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 dp-minimal expansions of the integers","authors":"Eran Alouf","doi":"10.1016/j.apal.2024.103551","DOIUrl":"10.1016/j.apal.2024.103551","url":null,"abstract":"<div><div>We show that if <span><math><mi>Z</mi></math></span> is a dp-minimal expansion of <span><math><mo>(</mo><mi>Z</mi><mo>,</mo><mo>+</mo><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span> that defines an infinite subset of <span><math><mi>N</mi></math></span>, then <span><math><mi>Z</mi></math></span> is interdefinable with <span><math><mo>(</mo><mi>Z</mi><mo>,</mo><mo>+</mo><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mo>&lt;</mo><mo>)</mo></math></span>. As a corollary, we show the same for dp-minimal expansions of <span><math><mo>(</mo><mi>Z</mi><mo>,</mo><mo>+</mo><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span> which do not eliminate <span><math><msup><mrow><mo>∃</mo></mrow><mrow><mo>∞</mo></mrow></msup></math></span>.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 4","pages":"Article 103551"},"PeriodicalIF":0.6,"publicationDate":"2025-01-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143162324","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":"Peano arithmetic, games and descent recursion","authors":"Emanuele Frittaion","doi":"10.1016/j.apal.2024.103550","DOIUrl":"10.1016/j.apal.2024.103550","url":null,"abstract":"<div><div>We analyze Coquand's game-theoretic interpretation of Peano Arithmetic <span><span>[6]</span></span> through the lens of elementary descent recursion <span><span>[8]</span></span>. In Coquand's game semantics, winning strategies correspond to infinitary cut-free proofs and cut elimination corresponds to <em>debates</em> between these winning strategies. The proof of cut elimination, i.e., the proof that such debates eventually terminate, is by transfinite induction on certain <em>interaction</em> sequences of ordinals. In this paper, we provide a direct implementation of Coquand's proof, one that allows us to describe winning strategies by descent recursive functions. As a byproduct, we obtain yet another proof of well-known results about provably recursive functions and functionals.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 4","pages":"Article 103550"},"PeriodicalIF":0.6,"publicationDate":"2024-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143162345","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":"The logic of cardinality comparison without the axiom of choice","authors":"Matthew Harrison-Trainor ,&nbsp;Dhruv Kulshreshtha","doi":"10.1016/j.apal.2024.103549","DOIUrl":"10.1016/j.apal.2024.103549","url":null,"abstract":"<div><div>We work in the setting of Zermelo-Fraenkel set theory without assuming the Axiom of Choice. We consider sets with the Boolean operations together with the additional structure of comparing cardinality (in the Cantorian sense of injections). What principles does one need to add to the laws of Boolean algebra to reason not only about intersection, union, and complementation of sets, but also about the relative size of sets? We give a complete axiomatization.</div><div>A particularly interesting case is when one restricts to the Dedekind-finite sets. In this case, one needs exactly the same principles as for reasoning about imprecise probability comparisons, the central principle being Generalized Finite Cancellation (which includes, as a special case, division-by-<em>m</em>). In the general case, the central principle is a restricted version of Generalized Finite Cancellation within Archimedean classes which we call Covered Generalized Finite Cancellation.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 4","pages":"Article 103549"},"PeriodicalIF":0.6,"publicationDate":"2024-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143162346","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":"Ordered transexponential fields","authors":"Lothar Sebastian Krapp ,&nbsp;Salma Kuhlmann","doi":"10.1016/j.apal.2024.103541","DOIUrl":"10.1016/j.apal.2024.103541","url":null,"abstract":"<div><div>We develop a first-order theory of ordered transexponential fields in the language <span><math><mo>{</mo><mo>+</mo><mo>,</mo><mo>⋅</mo><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mo>&lt;</mo><mo>,</mo><mi>e</mi><mo>,</mo><mi>T</mi><mo>}</mo></math></span>, where <em>e</em> and <em>T</em> stand for unary function symbols. While the archimedean models of this theory are readily described, the study of the non-archimedean models leads to a systematic examination of the induced structure on the residue field and the value group under the natural valuation. We establish necessary and sufficient conditions on the value group of an ordered exponential field <span><math><mo>(</mo><mi>K</mi><mo>,</mo><mi>e</mi><mo>)</mo></math></span> to admit a transexponential function <em>T</em> compatible with <em>e</em>. Moreover, we give a full characterisation of all <em>countable</em> ordered transexponential fields in terms of their valuation theoretic invariants.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 4","pages":"Article 103541"},"PeriodicalIF":0.6,"publicationDate":"2024-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143162350","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":"Tame topology in Hensel minimal structures","authors":"Krzysztof Jan Nowak","doi":"10.1016/j.apal.2024.103540","DOIUrl":"10.1016/j.apal.2024.103540","url":null,"abstract":"<div><div>We are concerned with topology of Hensel minimal structures on non-trivially valued fields <em>K</em>, whose axiomatic theory was introduced in a recent paper by Cluckers–Halupczok–Rideau. We additionally require that every definable subset in the imaginary sort <em>RV</em>, binding together the residue field <em>Kv</em> and value group <em>vK</em>, be already definable in the plain valued field language. This condition is satisfied by several classical tame structures on Henselian fields, including Henselian fields with analytic structure, V-minimal fields, and polynomially bounded o-minimal structures with a convex subring. In this article, we establish many results concerning definable functions and sets. These are, among others, existence of the limit for definable functions of one variable, a closedness theorem, several non-Archimedean versions of the Łojasiewicz inequalities, an embedding theorem for regular definable spaces, and the definable ultranormality and ultraparacompactness of definable Hausdorff LC-spaces.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 4","pages":"Article 103540"},"PeriodicalIF":0.6,"publicationDate":"2024-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143162351","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":"Strength and limitations of Sherali-Adams and Nullstellensatz proof systems","authors":"Ilario Bonacina,&nbsp;Maria Luisa Bonet","doi":"10.1016/j.apal.2024.103538","DOIUrl":"10.1016/j.apal.2024.103538","url":null,"abstract":"<div><div>We compare the strength of the algebraic proof systems Sherali-Adams (<span><math><mi>SA</mi></math></span>) and Nullstellensatz (<span><math><mi>NS</mi></math></span>) with Frege-style proof systems. Unlike bounded-depth Frege, <span><math><mi>SA</mi></math></span> has polynomial-size proofs of the pigeonhole principle (<span>PHP</span>). A natural question is whether adding <span>PHP</span> to bounded-depth Frege is enough to simulate <span><math><mi>SA</mi></math></span>. We show that <span><math><mi>SA</mi></math></span>, with unary integer coefficients, lies strictly between tree-like depth-1 <span><math><mtext>Frege</mtext><mo>+</mo><mrow><mi>PHP</mi></mrow></math></span> and tree-like <span><math><mtext>Resolution</mtext></math></span>. We introduce a <em>levelled</em> version of <span>PHP</span> (<span><math><mi>L</mi><mrow><mi>PHP</mi></mrow></math></span>) and we show that <span><math><mi>SA</mi></math></span> with integer coefficients lies strictly between tree-like depth-1 <span><math><mtext>Frege</mtext><mo>+</mo><mi>L</mi><mrow><mi>PHP</mi></mrow></math></span> and <span><math><mtext>Resolution</mtext></math></span>. Analogous results are shown for <span><math><mi>NS</mi></math></span> using the bijective (i.e. onto and functional) pigeonhole principle and a leveled version of it.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 4","pages":"Article 103538"},"PeriodicalIF":0.6,"publicationDate":"2024-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143162344","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":"More about the cofinality and the covering of the ideal of strong measure zero sets","authors":"Miguel A. Cardona ,&nbsp;Diego A. Mejía","doi":"10.1016/j.apal.2024.103537","DOIUrl":"10.1016/j.apal.2024.103537","url":null,"abstract":"<div><div>We improve the previous work of Yorioka and the first author about the combinatorics of the ideal <span><math><mi>SN</mi></math></span> of strong measure zero sets of reals. We refine the notions of dominating systems of the first author and introduce the new combinatorial principle <span><math><mrow><mi>DS</mi></mrow><mo>(</mo><mi>δ</mi><mo>)</mo></math></span> that helps to find simple conditions to deduce <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>κ</mi></mrow></msub><mo>≤</mo><mrow><mi>cof</mi></mrow><mo>(</mo><mrow><mi>SN</mi></mrow><mo>)</mo></math></span> (where <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>κ</mi></mrow></msub></math></span> is the dominating number on <span><math><msup><mrow><mi>κ</mi></mrow><mrow><mi>κ</mi></mrow></msup></math></span>). In addition, we find a new upper bound of <span><math><mrow><mi>cof</mi></mrow><mo>(</mo><mrow><mi>SN</mi></mrow><mo>)</mo></math></span> by using products of relational systems and cardinal characteristics associated with Yorioka ideals.</div><div>In addition, we dissect and generalize results from Pawlikowski to force upper bounds of the covering of <span><math><mi>SN</mi></math></span>, particularly for finite support iterations of precaliber posets.</div><div>Finally, as applications of our main theorems, we prove consistency results about the cardinal characteristics associated with <span><math><mi>SN</mi></math></span> and the principle <span><math><mrow><mi>DS</mi></mrow><mo>(</mo><mi>δ</mi><mo>)</mo></math></span>. For example, we show that <span><math><mrow><mi>cov</mi></mrow><mo>(</mo><mrow><mi>SN</mi></mrow><mo>)</mo><mo>&lt;</mo><mrow><mi>non</mi></mrow><mo>(</mo><mrow><mi>SN</mi></mrow><mo>)</mo><mo>=</mo><mi>c</mi><mo>&lt;</mo><mrow><mi>cof</mi></mrow><mo>(</mo><mrow><mi>SN</mi></mrow><mo>)</mo></math></span> holds in Cohen model, and we refine a result (and the proof) of the first author about the consistency of <span><math><mrow><mi>cov</mi></mrow><mo>(</mo><mrow><mi>SN</mi></mrow><mo>)</mo><mo>&lt;</mo><mrow><mi>non</mi></mrow><mo>(</mo><mrow><mi>SN</mi></mrow><mo>)</mo><mo>&lt;</mo><mrow><mi>cof</mi></mrow><mo>(</mo><mrow><mi>SN</mi></mrow><mo>)</mo></math></span>, with <span><math><mi>c</mi></math></span> in any desired position with respect to <span><math><mrow><mi>cof</mi></mrow><mo>(</mo><mrow><mi>SN</mi></mrow><mo>)</mo></math></span>, and the improvement that <span><math><mrow><mi>non</mi></mrow><mo>(</mo><mrow><mi>SN</mi></mrow><mo>)</mo></math></span> can be singular here.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 4","pages":"Article 103537"},"PeriodicalIF":0.6,"publicationDate":"2024-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143162352","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}