Annals of Pure and Applied Logic最新文献

{"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}

{"title":"Positive definability patterns","authors":"Ori Segel","doi":"10.1016/j.apal.2024.103539","DOIUrl":"10.1016/j.apal.2024.103539","url":null,"abstract":"<div><div>We reformulate Hrushovski's definability patterns from the setting of first order logic to the setting of positive logic. Given an h-universal theory <em>T</em> we put two structures on the type spaces of models of <em>T</em> in two languages, <span><math><mi>L</mi></math></span> and <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>π</mi></mrow></msub></math></span>. It turns out that for sufficiently saturated models, the corresponding h-universal theories <span><math><mi>T</mi></math></span> and <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>π</mi></mrow></msub></math></span> are independent of the model. We show that there is a canonical model <span><math><mi>J</mi></math></span> of <span><math><mi>T</mi></math></span>, and in many interesting cases there is an analogous canonical model <span><math><msub><mrow><mi>J</mi></mrow><mrow><mi>π</mi></mrow></msub></math></span> of <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>π</mi></mrow></msub></math></span>, both of which embed into every type space. We discuss the properties of these canonical models, called cores, and give some concrete examples.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 4","pages":"Article 103539"},"PeriodicalIF":0.6,"publicationDate":"2024-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143162349","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":"Strong standard completeness theorems for S5-modal Łukasiewicz logics","authors":"Diego Castaño ,&nbsp;José Patricio Díaz Varela ,&nbsp;Gabriel Savoy","doi":"10.1016/j.apal.2024.103529","DOIUrl":"10.1016/j.apal.2024.103529","url":null,"abstract":"<div><div>We study the S5-modal expansion of the Łukasiewicz logic. We exhibit a finitary propositional calculus and show that it is finitely strongly complete with respect to this logic. This propositional calculus is then expanded with an infinitary rule to achieve strong completeness. These results are derived from properties of monadic MV-algebras: functional representations of simple and finitely subdirectly irreducible algebras, as well as the finite embeddability property. We also show similar completeness theorems for the extension of the logic based on models with bounded universe.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 3","pages":"Article 103529"},"PeriodicalIF":0.6,"publicationDate":"2024-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142746012","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":"Semiconic idempotent logic II: Beth definability and deductive interpolation","authors":"Wesley Fussner ,&nbsp;Nikolaos Galatos","doi":"10.1016/j.apal.2024.103528","DOIUrl":"10.1016/j.apal.2024.103528","url":null,"abstract":"<div><div>Semiconic idempotent logic <strong>sCI</strong> is a common generalization of intuitionistic logic, semilinear idempotent logic <strong>sLI</strong>, and in particular relevance logic with mingle. We establish the projective Beth definability property and the deductive interpolation property for many extensions of <strong>sCI</strong>, and identify extensions where these properties fail. We achieve these results by studying the (strong) amalgamation property and the epimorphism-surjectivity property for the corresponding algebraic semantics, viz. semiconic idempotent residuated lattices. Our study is made possible by the structural decomposition of conic idempotent models achieved in the prequel, as well as a detailed analysis of the structure of idempotent residuated chains serving as index sets in this decomposition. Here we study the latter on two levels: as certain enriched Galois connections and as enhanced monoidal preorders. Using this, we show that although conic idempotent residuated lattices do not have the amalgamation property, the natural class of stratified and conjunctive conic idempotent residuated lattices has the strong amalgamation property, and thus has surjective epimorphisms. This extends to the variety generated by stratified and conjunctive conic idempotent residuated lattices, and we establish the (strong) amalgamation and epimorphism-surjectivity properties for several important subvarieties. Using the algebraizability of <strong>sCI</strong>, this yields the deductive interpolation property and the projective Beth definability property for the corresponding substructural logics extending <strong>sCI</strong>.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 3","pages":"Article 103528"},"PeriodicalIF":0.6,"publicationDate":"2024-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143148056","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":"Failure of the Blok–Esakia Theorem in the monadic setting","authors":"G. Bezhanishvili ,&nbsp;L. Carai","doi":"10.1016/j.apal.2024.103527","DOIUrl":"10.1016/j.apal.2024.103527","url":null,"abstract":"<div><div>The Blok–Esakia Theorem establishes that the lattice of superintuitionistic logics is isomorphic to the lattice of extensions of Grzegorczyk's logic. We prove that the Blok–Esakia isomorphism <em>σ</em> does not extend to the fragments of the corresponding predicate logics of already one fixed variable. In other words, we prove that <em>σ</em> is no longer an isomorphism from the lattice of extensions of the monadic intuitionistic logic to the lattice of extensions of the monadic Grzegorczyk logic.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 4","pages":"Article 103527"},"PeriodicalIF":0.6,"publicationDate":"2024-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143162353","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":"Universal proof theory: Feasible admissibility in intuitionistic modal logics","authors":"Amirhossein Akbar Tabatabai ,&nbsp;Raheleh Jalali","doi":"10.1016/j.apal.2024.103526","DOIUrl":"10.1016/j.apal.2024.103526","url":null,"abstract":"<div><div>We introduce a general and syntactically defined family of sequent-style calculi over the propositional language with the modalities <span><math><mo>{</mo><mo>□</mo><mo>,</mo><mo>◇</mo><mo>}</mo></math></span> and its fragments as a formalization for constructively acceptable systems. Calling these calculi <em>constructive</em>, we show that any strong enough constructive sequent calculus, satisfying a mild technical condition, feasibly admits all Visser's rules. This means that there exists a polynomial-time algorithm that, given a proof of the premise of a Visser's rule, provides a proof for its conclusion. As a positive application, we establish the feasible admissibility of Visser's rules in sequent calculi for several intuitionistic modal logics, including <span><math><mi>CK</mi></math></span>, <span><math><mi>IK</mi></math></span>, their extensions by the modal axioms <em>T</em>, <em>B</em>, 4, 5, and the axioms for bounded width and depth and their fragments <span><math><msub><mrow><mi>CK</mi></mrow><mrow><mo>□</mo></mrow></msub></math></span>, propositional lax logic and <span><math><mi>IPC</mi></math></span>. On the negative side, we show that if a strong enough intuitionistic modal logic (satisfying a mild technical condition) does not admit at least one of Visser's rules, it cannot have a constructive sequent calculus. Consequently, no intermediate logic other than <span><math><mi>IPC</mi></math></span> has a constructive sequent calculus.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 2","pages":"Article 103526"},"PeriodicalIF":0.6,"publicationDate":"2024-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142561348","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}