Annals of Pure and Applied Logic最新文献

{"title":"On cardinal invariants related to Rosenthal families and large-scale topology","authors":"Arturo Martínez-Celis,&nbsp;Tomasz Żuchowski","doi":"10.1016/j.apal.2025.103607","DOIUrl":"10.1016/j.apal.2025.103607","url":null,"abstract":"<div><div>Given a function <span><math><mi>f</mi><mo>∈</mo><msup><mrow><mi>ω</mi></mrow><mrow><mi>ω</mi></mrow></msup></math></span>, a set <span><math><mi>A</mi><mo>∈</mo><msup><mrow><mo>[</mo><mi>ω</mi><mo>]</mo></mrow><mrow><mi>ω</mi></mrow></msup></math></span> is <em>free for f</em> if <span><math><mi>f</mi><mo>[</mo><mi>A</mi><mo>]</mo><mo>∩</mo><mi>A</mi></math></span> is finite. For a class of functions <span><math><mi>Γ</mi><mo>⊆</mo><msup><mrow><mi>ω</mi></mrow><mrow><mi>ω</mi></mrow></msup></math></span>, we define <span><math><msub><mrow><mi>ros</mi></mrow><mrow><mi>Γ</mi></mrow></msub></math></span> as the smallest size of a family <span><math><mi>A</mi><mo>⊆</mo><msup><mrow><mo>[</mo><mi>ω</mi><mo>]</mo></mrow><mrow><mi>ω</mi></mrow></msup></math></span> such that for every <span><math><mi>f</mi><mo>∈</mo><mi>Γ</mi></math></span> there is a set <span><math><mi>A</mi><mo>∈</mo><mi>A</mi></math></span> which is free for <em>f</em>, and <span><math><msub><mrow><mi>Δ</mi></mrow><mrow><mi>Γ</mi></mrow></msub></math></span> as the smallest size of a family <span><math><mi>F</mi><mo>⊆</mo><mi>Γ</mi></math></span> such that for every <span><math><mi>A</mi><mo>∈</mo><msup><mrow><mo>[</mo><mi>ω</mi><mo>]</mo></mrow><mrow><mi>ω</mi></mrow></msup></math></span> there is <span><math><mi>f</mi><mo>∈</mo><mi>F</mi></math></span> such that <em>A</em> is not free for <em>f</em>. We compare several versions of these cardinal invariants with some of the classical cardinal characteristics of the continuum. Using these notions, we partially answer some questions from <span><span>[20]</span></span> and <span><span>[2]</span></span>.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 8","pages":"Article 103607"},"PeriodicalIF":0.6,"publicationDate":"2025-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143922505","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":"Generically computable linear orderings","authors":"Wesley Calvert ,&nbsp;Douglas Cenzer ,&nbsp;David Gonzalez ,&nbsp;Valentina Harizanov","doi":"10.1016/j.apal.2025.103612","DOIUrl":"10.1016/j.apal.2025.103612","url":null,"abstract":"<div><div>We study notions of generic and coarse computability in the context of computable structure theory. Our notions are stratified by the <span><math><msub><mrow><mi>Σ</mi></mrow><mrow><mi>β</mi></mrow></msub></math></span> hierarchy. We focus on linear orderings. We show that at the <span><math><msub><mrow><mi>Σ</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> level, all linear orderings have both generically and coarsely computable copies. This behavior changes abruptly at higher levels; we show that at the <span><math><msub><mrow><mi>Σ</mi></mrow><mrow><mi>α</mi><mo>+</mo><mn>2</mn></mrow></msub></math></span> level for any <span><math><mi>α</mi><mo>∈</mo><msubsup><mrow><mi>ω</mi></mrow><mrow><mn>1</mn></mrow><mrow><mi>C</mi><mi>K</mi></mrow></msubsup></math></span> the set of linear orderings with generically or coarsely computable copies is <span><math><msubsup><mrow><mi>Σ</mi></mrow><mrow><mn>1</mn></mrow><mrow><mn>1</mn></mrow></msubsup></math></span>-complete and therefore maximally complicated. This development is new even in the general analysis of generic and coarse computability of countable structures. In the process of proving these results, we introduce new tools for understanding generically and coarsely computable structures. We are able to give a purely structural statement that is equivalent to having a generically computable copy and show that every relational structure with only finitely many relations has coarsely and generically computable copies at the lowest level of the hierarchy.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 8","pages":"Article 103612"},"PeriodicalIF":0.6,"publicationDate":"2025-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143943554","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":"Spaces not distinguishing ideal pointwise and σ-uniform convergence","authors":"Rafał Filipów,&nbsp;Adam Kwela","doi":"10.1016/j.apal.2025.103609","DOIUrl":"10.1016/j.apal.2025.103609","url":null,"abstract":"<div><div>We examine topological spaces not distinguishing ideal pointwise and ideal <em>σ</em>-uniform convergence of sequences of real-valued continuous functions defined on them. For instance, we introduce a purely combinatorial cardinal characteristic (a sort of the bounding number <span><math><mi>b</mi></math></span>) and prove that it describes the minimal cardinality of topological spaces which distinguish ideal pointwise and ideal <em>σ</em>-uniform convergence. Moreover, we provide examples of topological spaces (focusing on subsets of reals) that do or do not distinguish the considered convergences. Since similar investigations for ideal quasi-normal convergence instead of ideal <em>σ</em>-uniform convergence have been performed in literature, we also study spaces not distinguishing ideal quasi-normal and ideal <em>σ</em>-uniform convergence of sequences of real-valued continuous functions defined on them.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 9","pages":"Article 103609"},"PeriodicalIF":0.6,"publicationDate":"2025-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143943439","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":"Metric spaces in choiceless set theory","authors":"Eleftherios Tachtsis","doi":"10.1016/j.apal.2025.103603","DOIUrl":"10.1016/j.apal.2025.103603","url":null,"abstract":"&lt;div&gt;&lt;div&gt;We &lt;em&gt;answer open questions&lt;/em&gt; from Keremedis (2016) &lt;span&gt;&lt;span&gt;[12]&lt;/span&gt;&lt;/span&gt; and Keremedis and Tachtsis (2022) &lt;span&gt;&lt;span&gt;[16]&lt;/span&gt;&lt;/span&gt;, and &lt;em&gt;properly strengthen some results&lt;/em&gt; from the above papers as well as from Keremedis et al. (2023) &lt;span&gt;&lt;span&gt;[19]&lt;/span&gt;&lt;/span&gt;. In particular, and among other results, we establish the following:&lt;ul&gt;&lt;li&gt;&lt;span&gt;1.&lt;/span&gt;&lt;span&gt;&lt;div&gt;The Boolean Prime Ideal Theorem does not imply “For every sequentially compact metric space &lt;span&gt;&lt;math&gt;&lt;mo&gt;〈&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mo&gt;〉&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;ℵ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;” in &lt;strong&gt;ZF&lt;/strong&gt; (Zermelo–Fraenkel set theory without the Axiom of Choice (&lt;strong&gt;AC&lt;/strong&gt;)).&lt;/div&gt;&lt;/span&gt;&lt;/li&gt;&lt;li&gt;&lt;span&gt;2.&lt;/span&gt;&lt;span&gt;&lt;div&gt;“Every linearly ordered set can be well ordered” ∧ “The union of a well-orderable family of well-orderable sets is well orderable” ∧ “For every uncountable sequentially compact metric space &lt;span&gt;&lt;math&gt;&lt;mo&gt;〈&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mo&gt;〉&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;ℵ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;” does not imply the Axiom of Countable Choice in &lt;strong&gt;ZFA&lt;/strong&gt; (&lt;strong&gt;ZF&lt;/strong&gt; with atoms).&lt;/div&gt;&lt;/span&gt;&lt;/li&gt;&lt;li&gt;&lt;span&gt;3.&lt;/span&gt;&lt;span&gt;&lt;div&gt;“For every uncountable compact metric space &lt;span&gt;&lt;math&gt;&lt;mo&gt;〈&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mo&gt;〉&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;ℵ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;” does not imply “For every uncountable sequentially compact metric space &lt;span&gt;&lt;math&gt;&lt;mo&gt;〈&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mo&gt;〉&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;ℵ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;” in &lt;strong&gt;ZFA&lt;/strong&gt;”.&lt;/div&gt;&lt;/span&gt;&lt;/li&gt;&lt;li&gt;&lt;span&gt;4.&lt;/span&gt;&lt;span&gt;&lt;div&gt;“For every uncountable sequentially compact metric space &lt;span&gt;&lt;math&gt;&lt;mo&gt;〈&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mo&gt;〉&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; &lt;span&gt;&lt;math&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;ℵ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;” does not imply “For every uncountable compact metric space &lt;span&gt;&lt;math&gt;&lt;mo&gt;〈&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mo&gt;〉&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;ℵ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;” in &lt;strong&gt;ZFA&lt;/strong&gt;.&lt;/div&gt;&lt;/span&gt;&lt;/li&gt;&lt;li&gt;&lt;span&gt;5.&lt;/span&gt;&lt;span&gt;&lt;div&gt;“For every uncountable sequentially compact metric space &lt;span&gt;&lt;math&gt;&lt;mo&gt;〈&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mo&gt;","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 9","pages":"Article 103603"},"PeriodicalIF":0.6,"publicationDate":"2025-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143886674","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":"Weakly o-minimal types","authors":"Slavko Moconja ,&nbsp;Predrag Tanović","doi":"10.1016/j.apal.2025.103605","DOIUrl":"10.1016/j.apal.2025.103605","url":null,"abstract":"<div><div>We introduce and study weak o-minimality in the context of complete types in an arbitrary first-order theory. A type <span><math><mi>p</mi><mo>∈</mo><mi>S</mi><mo>(</mo><mi>A</mi><mo>)</mo></math></span> is weakly o-minimal if for some relatively <em>A</em>-definable linear order, &lt;, on <span><math><mi>p</mi><mo>(</mo><mi>C</mi><mo>)</mo></math></span> every relatively <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>C</mi></mrow></msub></math></span>-definable subset of <span><math><mi>p</mi><mo>(</mo><mi>C</mi><mo>)</mo></math></span> has finitely many convex components in <span><math><mo>(</mo><mi>p</mi><mo>(</mo><mi>C</mi><mo>)</mo><mo>,</mo><mo>&lt;</mo><mo>)</mo></math></span>. We establish many nice properties of weakly o-minimal types. For example, we prove that weakly o-minimal types are dp-minimal and share several properties of weight-one types in stable theories, and that a version of monotonicity theorem holds for relatively definable functions on the locus of a weakly o-minimal type.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 9","pages":"Article 103605"},"PeriodicalIF":0.6,"publicationDate":"2025-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143922216","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":"Upward Löwenheim-Skolem-Tarski numbers for abstract logics","authors":"Victoria Gitman ,&nbsp;Jonathan Osinski","doi":"10.1016/j.apal.2025.103583","DOIUrl":"10.1016/j.apal.2025.103583","url":null,"abstract":"<div><div>Galeotti, Khomskii and Väänänen recently introduced the notion of the upward Löwenheim-Skolem-Tarski number for a logic, strengthening the classical notion of a Hanf number. A cardinal <em>κ</em> is the <em>upward Löwenheim-Skolem-Tarski number</em> (ULST <em>number</em>) of a logic <span><math><mi>L</mi></math></span> if it is the least cardinal with the property that whenever <em>M</em> is a model of size at least <em>κ</em> satisfying a sentence <em>φ</em> in <span><math><mi>L</mi></math></span>, then there are arbitrarily large models satisfying <em>φ</em> and having <em>M</em> as a substructure. The substructure requirement is what differentiates the ULST number from the Hanf number and gives the notion large cardinal strength. While it is a theorem of ZFC that every logic has a Hanf number, Galeotti, Khomskii and Väänänen showed that the existence of the ULST number for second-order logic implies the existence of a partially extendible cardinal. We answer positively their conjecture that the ULST number for second-order logic is the least extendible cardinal.</div><div>We define the <em>strong</em> ULST number by strengthening the substructure requirement to elementary substructure. We investigate the ULST and strong ULST numbers for several classical strong logics: infinitary logics, the equicardinality logic, logic with the well-foundedness quantifier, second-order logic, and sort logics. We show that the ULST and the strong ULST numbers are characterized in some cases by classical large cardinals and in some cases by natural new large cardinal notions that they give rise to. We show that for some logics the notions of the ULST number, strong ULST number and least strong compactness cardinal coincide, while for others, it is consistent that they can be separated. Finally, we introduce a natural large cardinal notion characterizing strong compactness cardinals for the equicardinality logic.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 8","pages":"Article 103583"},"PeriodicalIF":0.6,"publicationDate":"2025-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143748012","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 power of the Binary Value Principle","authors":"Yaroslav Alekseev ,&nbsp;Edward A. Hirsch","doi":"10.1016/j.apal.2025.103614","DOIUrl":"10.1016/j.apal.2025.103614","url":null,"abstract":"<div><div>The (extended) Binary Value Principle (<span><math><mi>eBVP</mi></math></span>, the equation <span><math><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></msubsup><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><msup><mrow><mn>2</mn></mrow><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>=</mo><mo>−</mo><mi>k</mi></math></span> for <span><math><mi>k</mi><mo>&gt;</mo><mn>0</mn></math></span> and Boolean variables <span><math><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span>) has received a lot of attention recently, several lower bounds have been proved for it <span><span>[1]</span></span>, <span><span>[2]</span></span>, <span><span>[11]</span></span>. Also it has been shown <span><span>[1]</span></span> that the probabilistically verifiable Ideal Proof System (<span><math><mi>IPS</mi></math></span>) <span><span>[8]</span></span> together with <span><math><mi>eBVP</mi></math></span> polynomially simulates a similar semialgebraic proof system. In this paper we consider Polynomial Calculus with an algebraic version of Tseitin's extension rule (<span><math><mrow><mi>Ext</mi></mrow><mtext>-</mtext><mrow><mi>PC</mi></mrow></math></span>) that introduces a new variable for any polynomial. Contrary to <span><math><mi>IPS</mi></math></span>, this is a Cook–Reckhow proof system. We show that in this context <span><math><mi>eBVP</mi></math></span> still allows to simulate similar semialgebraic systems. We also prove that it allows to simulate the Square Root Rule <span><span>[6]</span></span>, which is in sharp contrast with the result of <span><span>[2]</span></span> that shows an exponential lower bound on the size of <span><math><mrow><mi>Ext</mi></mrow><mtext>-</mtext><mrow><mi>PC</mi></mrow></math></span> derivations of the Binary Value Principle from its square. On the other hand, we demonstrate that <span><math><mi>eBVP</mi></math></span> probably does not help in proving exponential lower bounds for Boolean formulas: we show that an <span><math><mrow><mi>Ext</mi></mrow><mtext>-</mtext><mrow><mi>PC</mi></mrow></math></span> (even with the Square Root Rule) derivation of any unsatisfiable Boolean formula in CNF from <span><math><mi>eBVP</mi></math></span> must be of exponential size.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 9","pages":"Article 103614"},"PeriodicalIF":0.6,"publicationDate":"2025-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144195892","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":"The singleton degrees of the Σ20 sets are not dense","authors":"Thomas F. Kent ,&nbsp;Keng Meng Ng ,&nbsp;Andrea Sorbi","doi":"10.1016/j.apal.2025.103616","DOIUrl":"10.1016/j.apal.2025.103616","url":null,"abstract":"<div><div>Answering an open question raised by Cooper, we show that there exist <span><math><msubsup><mrow><mi>Δ</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>0</mn></mrow></msubsup></math></span> sets <em>D</em> and <em>E</em> such that the singleton degree of <em>E</em> is a minimal cover of the singleton degree of <em>D</em>. This shows that the <span><math><msubsup><mrow><mi>Σ</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>0</mn></mrow></msubsup></math></span> singleton degrees, and the <span><math><msubsup><mrow><mi>Δ</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>0</mn></mrow></msubsup></math></span> singleton degrees, are not dense (and consequently the <span><math><msubsup><mrow><mi>Π</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>0</mn></mrow></msubsup></math></span> <em>Q</em>-degrees, and the <span><math><msubsup><mrow><mi>Δ</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>0</mn></mrow></msubsup></math></span> <em>Q</em>-degrees, are not dense). Moreover, <em>D</em> and <em>E</em> can be built to lie in the same enumeration degree.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 9","pages":"Article 103616"},"PeriodicalIF":0.6,"publicationDate":"2025-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144195893","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":"Free p-algebras revisited: An algebraic investigation of implication-free intuitionism","authors":"Tomasz Kowalski ,&nbsp;Katarzyna Słomczyńska","doi":"10.1016/j.apal.2025.103610","DOIUrl":"10.1016/j.apal.2025.103610","url":null,"abstract":"<div><div>We give a new construction of free distributive p-algebras. Our construction relies on a detailed description of completely meet-irreducible congruences, so it is purely universal algebraic. It yields a normal form theorem for p-algebra terms, simpler proofs of several existing results. As a by-product, we obtain an isomorphism between the free pseudocomplemented semilattice and the poset of join-irreducibles of the free p-algebra augmented by zero.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 9","pages":"Article 103610"},"PeriodicalIF":0.6,"publicationDate":"2025-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143943440","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":"Model-theoretic K1 of free modules over PIDs","authors":"Sourayan Banerjee,&nbsp;Amit Kuber","doi":"10.1016/j.apal.2025.103613","DOIUrl":"10.1016/j.apal.2025.103613","url":null,"abstract":"<div><div>Motivated by Krajiček and Scanlon's definition of the Grothendieck ring <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>M</mi><mo>)</mo></math></span> of a first-order structure <em>M</em>, we introduce the definition of <em>K</em>-groups <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>M</mi><mo>)</mo></math></span> for <span><math><mi>n</mi><mo>≥</mo><mn>0</mn></math></span> via Quillen's <span><math><msup><mrow><mi>S</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mi>S</mi></math></span> construction. We provide a recipe for the computation of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>R</mi></mrow></msub><mo>)</mo></math></span>, where <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>R</mi></mrow></msub></math></span> is a free module over a PID <em>R</em>, subject to the knowledge of the abelianizations of the general linear groups <span><math><mi>G</mi><msub><mrow><mi>L</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>R</mi><mo>)</mo></math></span>. As a consequence, we provide explicit computations of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>R</mi></mrow></msub><mo>)</mo></math></span> when <em>R</em> belongs to a large class of Euclidean domains that includes fields with at least 3 elements and polynomial rings over fields with characteristic 0. We also show that the algebraic <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> of a PID <em>R</em> embeds into <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><msub><mrow><mi>R</mi></mrow><mrow><mi>R</mi></mrow></msub><mo>)</mo></math></span>.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 9","pages":"Article 103613"},"PeriodicalIF":0.6,"publicationDate":"2025-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144116947","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}