{"title":"SOP1, SOP2, and antichain tree property","authors":"JinHoo Ahn , Joonhee Kim","doi":"10.1016/j.apal.2023.103402","DOIUrl":"10.1016/j.apal.2023.103402","url":null,"abstract":"<div><p>In this paper, we study some tree properties and their related indiscernibilities. First, we prove that SOP<sub>2</sub> can be witnessed by a formula with a tree of tuples holding ‘arbitrary homogeneous inconsistency’ (e.g., weak <em>k</em>-TP<sub>1</sub> conditions or other possible inconsistency configurations).</p><p>And we introduce a notion of tree-indiscernibility, which preserves witnesses of SOP<sub>1</sub>, and by using this, we investigate the problem of (in)equality of SOP<sub>1</sub> and SOP<sub>2</sub>.</p><p>Assuming the existence of a formula having SOP<sub>1</sub> such that no finite conjunction of it has SOP<sub>2</sub><span>, we observe that the formula must witness some tree-property-like phenomenon, which we will call the antichain tree property (ATP, see </span><span>Definition 4.1</span>). We show that ATP implies SOP<sub>1</sub> and TP<sub>2</sub>, but the converse of each implication does not hold. So the class of NATP theories (theories without ATP) contains the class of NSOP<sub>1</sub> theories and the class of NTP<sub>2</sub> theories.</p><p>At the end of the paper, we construct a structure whose theory has a formula having ATP, but any conjunction of the formula does not have SOP<sub>2</sub>. So this example shows that SOP<sub>1</sub> and SOP<sub>2</sub> are not the same at the level of formulas, i.e., there is a formula having SOP<sub>1</sub>, while any finite conjunction of it does not witness SOP<sub>2</sub> (but a variation of the formula still has SOP<sub>2</sub>).</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"175 3","pages":"Article 103402"},"PeriodicalIF":0.8,"publicationDate":"2023-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138575042","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":"Constructing the constructible universe constructively","authors":"Richard Matthews , Michael Rathjen","doi":"10.1016/j.apal.2023.103392","DOIUrl":"10.1016/j.apal.2023.103392","url":null,"abstract":"<div><p>We study the properties of the constructible universe, L, over intuitionistic theories. We give an extended set of fundamental operations which is sufficient to generate the universe over Intuitionistic Kripke-Platek set theory without Infinity. Following this, we investigate when L can fail to be an inner model in the traditional sense. Namely, we show that over Constructive Zermelo-Fraenkel (even with the Power Set axiom) one cannot prove that the Axiom of Exponentiation holds in L.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"175 3","pages":"Article 103392"},"PeriodicalIF":0.8,"publicationDate":"2023-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138548019","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 categorical structures arising from implicative algebras: From topology to assemblies","authors":"Samuele Maschio, Davide Trotta","doi":"10.1016/j.apal.2023.103390","DOIUrl":"https://doi.org/10.1016/j.apal.2023.103390","url":null,"abstract":"<div><p>Implicative algebras have been recently introduced by Miquel in order to provide a unifying notion of model, encompassing the most relevant and used ones, such as realizability (both classical and intuitionistic), and forcing. In this work, we initially approach implicative algebras as a generalization of locales, and we extend several topological-like concepts to the realm of implicative algebras, accompanied by various concrete examples. Then, we shift our focus to viewing implicative algebras as a generalization of partial combinatory algebras. We abstract the notion of a category of assemblies, partition assemblies, and modest sets to arbitrary implicative algebras, and thoroughly investigate their categorical properties and interrelationships.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"175 3","pages":"Article 103390"},"PeriodicalIF":0.8,"publicationDate":"2023-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0168007223001471/pdfft?md5=e62dfd1384a82cf70c38bd805eb49847&pid=1-s2.0-S0168007223001471-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138484884","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":"Probabilistic temporal logic with countably additive semantics","authors":"Dragan Doder , Zoran Ognjanović","doi":"10.1016/j.apal.2023.103389","DOIUrl":"10.1016/j.apal.2023.103389","url":null,"abstract":"<div><p>This work presents a proof-theoretical and model-theoretical approach to probabilistic temporal logic. We present two novel logics; each of them extends both the language of linear time logic (LTL) and the language of probabilistic logic with polynomial weight formulas. The first logic is designed for reasoning about probabilities of temporal events, allowing statements like “the probability that A will hold in next moment is at least the probability that B will always hold” and conditional probability statements like “probability that A will always hold, given that B holds, is at least one half”, where A and B are arbitrary statements. We axiomatize this logic, provide corresponding sigma additive semantics and prove that the axiomatization is sound and strongly complete. We show that the satisfiability problem for our logic is decidable, by presenting a procedure which runs in polynomial space. We also present a logic with much richer language, in which probabilities are not attached only to temporal events, but the language allows arbitrary nesting of probability and temporal operators, allowing statements like “probability that tomorrow the chance of rain will be less than 80% is at least a half”. For this logic we prove a decidability result.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"175 9","pages":"Article 103389"},"PeriodicalIF":0.8,"publicationDate":"2023-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S016800722300146X/pdfft?md5=b48bdc121fa115161627545db4c861fd&pid=1-s2.0-S016800722300146X-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135716247","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":"On duality and model theory for polyadic spaces","authors":"Sam van Gool, Jérémie Marquès","doi":"10.1016/j.apal.2023.103388","DOIUrl":"https://doi.org/10.1016/j.apal.2023.103388","url":null,"abstract":"<div><p>This paper is a study of first-order coherent logic from the point of view of duality and categorical logic. We prove a duality theorem between coherent hyperdoctrines and open polyadic Priestley spaces, which we subsequently apply to prove completeness, omitting types, and Craig interpolation theorems for coherent or intuitionistic logic. Our approach emphasizes the role of interpolation and openness properties, and allows for a modular, syntax-free treatment of these model-theoretic results. As further applications of the same method, we prove completeness theorems for constant domain and Gödel-Dummett intuitionistic predicate logics.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"175 2","pages":"Article 103388"},"PeriodicalIF":0.8,"publicationDate":"2023-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134656886","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":"Pathologies in satisfaction classes","authors":"Athar Abdul-Quader , Mateusz Łełyk","doi":"10.1016/j.apal.2023.103387","DOIUrl":"https://doi.org/10.1016/j.apal.2023.103387","url":null,"abstract":"<div><p>We study subsets of countable recursively saturated models of <span><math><mi>PA</mi></math></span> which can be defined using pathologies in satisfaction classes. More precisely, we characterize those subsets <em>X</em> such that there is a satisfaction class <em>S</em> where <em>S</em> behaves correctly on an idempotent disjunction of length <em>c</em> if and only if <span><math><mi>c</mi><mo>∈</mo><mi>X</mi></math></span>. We generalize this result to characterize several types of pathologies including double negations, blocks of extraneous quantifiers, and binary disjunctions and conjunctions. We find a surprising relationship between the cuts which can be defined in this way and arithmetic saturation: namely, a countable nonstandard model is arithmetically saturated if and only if every cut can be the “idempotent disjunctively correct cut” in some satisfaction class. We describe the relationship between types of pathologies and the closure properties of the cuts defined by these pathologies.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"175 2","pages":"Article 103387"},"PeriodicalIF":0.8,"publicationDate":"2023-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49725035","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 the geometric equivalence of algebras","authors":"M. Shahryari","doi":"10.1016/j.apal.2023.103386","DOIUrl":"https://doi.org/10.1016/j.apal.2023.103386","url":null,"abstract":"<div><p>It is known that an algebra is geometrically equivalent to any of its filterpowers if it is <span><math><msub><mrow><mi>q</mi></mrow><mrow><mi>ω</mi></mrow></msub></math></span>-compact. We present an explicit description for the radicals of systems of equation over an algebra <em>A</em> and then we prove the above assertion by an elementary new argument. Then we define <span><math><msub><mrow><mi>q</mi></mrow><mrow><mi>κ</mi></mrow></msub></math></span>-compact algebras and <em>κ</em>-filterpowers for any infinite cardinal <em>κ</em>. We show that any <span><math><msub><mrow><mi>q</mi></mrow><mrow><mi>κ</mi></mrow></msub></math></span>-compact algebra is geometric equivalent to its <em>κ</em>-filterpowers. As there is no algebraic description of the <em>κ</em>-quasivariety generated by an algebra, the classical argument can not be applied in this case, while our proof still works.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"175 2","pages":"Article 103386"},"PeriodicalIF":0.8,"publicationDate":"2023-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49725032","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":"Social welfare relations and irregular sets","authors":"Ram Sewak Dubey , Giorgio Laguzzi","doi":"10.1016/j.apal.2023.103302","DOIUrl":"https://doi.org/10.1016/j.apal.2023.103302","url":null,"abstract":"<div><p>Social welfare relations satisfying Pareto and equity principles on infinite utility streams have revealed a non-constructive nature, specifically by showing that in general they imply the existence of non-Ramsey sets and non-Lebesgue measurable sets. In <span>[4, Problem 11.14]</span>, the authors ask whether such a connection holds with non-Baire sets as well. In this paper we answer such a question showing that several versions of Pareto principles acting on different utility domains imply the existence of non-Baire sets. Furthermore, we analyze in more details the needed fragments of AC and we start a systematic investigation of a <em>social welfare diagram</em> in a similar fashion done in the past decades concerning cardinal invariants and regularity properties of the reals. In doing that we use tools from forcing theory, such as specific tree-forcings (in particular variants of Silver and Mathias forcings) and Shelah's amalgamation.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"174 9","pages":"Article 103302"},"PeriodicalIF":0.8,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49754694","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":"Krull dimension in set theory","authors":"Jindřich Zapletal","doi":"10.1016/j.apal.2023.103299","DOIUrl":"https://doi.org/10.1016/j.apal.2023.103299","url":null,"abstract":"<div><p>For every number <span><math><mi>n</mi><mo>≥</mo><mn>2</mn></math></span>, let <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> be the hypergraph on <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> of arity four consisting of all non-degenerate Euclidean rectangles. It is consistent with ZF+DC set theory that the chromatic number of <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> is countable while that of <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span> is not.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"174 9","pages":"Article 103299"},"PeriodicalIF":0.8,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49754695","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}
Paul-Elliot Angles d'Auriac , Lu Liu , Bastien Mignoty , Ludovic Patey
{"title":"Carlson-Simpson's lemma and applications in reverse mathematics","authors":"Paul-Elliot Angles d'Auriac , Lu Liu , Bastien Mignoty , Ludovic Patey","doi":"10.1016/j.apal.2023.103287","DOIUrl":"https://doi.org/10.1016/j.apal.2023.103287","url":null,"abstract":"<div><p>We study the reverse mathematics of infinitary extensions of the Hales-Jewett theorem, due to Carlson and Simpson. These theorems have multiple applications in Ramsey's theory, such as the existence of finite big Ramsey degrees for the triangle-free graph, or the Dual Ramsey theorem. We show in particular that the Open Dual Ramsey theorem holds in <span><math><msubsup><mrow><mi>ACA</mi></mrow><mrow><mn>0</mn></mrow><mrow><mo>+</mo></mrow></msubsup></math></span>.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"174 9","pages":"Article 103287"},"PeriodicalIF":0.8,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49754698","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}