Annals of Pure and Applied Logic最新文献

{"title":"ZF and its interpretations","authors":"S. Jockwich Martinez ,&nbsp;S. Tarafder ,&nbsp;G. Venturi","doi":"10.1016/j.apal.2024.103427","DOIUrl":"10.1016/j.apal.2024.103427","url":null,"abstract":"<div><p>In this paper, we unify the study of classical and non-classical algebra-valued models of set theory, by studying variations of the interpretation functions for = and ∈. Although, these variations coincide with the standard interpretation in Boolean-valued constructions, nonetheless they extend the scope of validity of <span><math><mi>ZF</mi></math></span> to new algebra-valued models. This paper presents, for the first time, non-trivial paraconsistent models of full <span><math><mi>ZF</mi></math></span>. Moreover, due to the validity of Leibniz's law in these structures, we will show how to construct proper models of set theory by quotienting these algebra-valued models with respect to equality, modulo the filter of the designated truth-values.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"175 6","pages":"Article 103427"},"PeriodicalIF":0.8,"publicationDate":"2024-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0168007224000241/pdfft?md5=9d7bf0eef51dc942a71051fb8dfcc3b5&pid=1-s2.0-S0168007224000241-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140054131","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":"A good lightface Δn1 well-ordering of the reals does not imply the existence of boldface Δn−11 well-orderings","authors":"Vladimir Kanovei,&nbsp;Vassily Lyubetsky","doi":"10.1016/j.apal.2024.103426","DOIUrl":"10.1016/j.apal.2024.103426","url":null,"abstract":"<div><p>We make use of a finite support product of the Jensen-type forcing notions to define a model of the set theory <span><math><mtext>ZFC</mtext></math></span> in which, for a given <span><math><mi>n</mi><mo>≥</mo><mn>3</mn></math></span>, there exists a good lightface <span><math><msubsup><mrow><mi>Δ</mi></mrow><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msubsup></math></span> well-ordering of the reals but there are no any (not necessarily good) well-orderings in the boldface class <span><math><msubsup><mrow><mi>Δ</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mrow><mn>1</mn></mrow></msubsup></math></span>.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"175 6","pages":"Article 103426"},"PeriodicalIF":0.8,"publicationDate":"2024-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140054039","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":"Nonstandard proof methods in toposes","authors":"José Siqueira","doi":"10.1016/j.apal.2024.103424","DOIUrl":"10.1016/j.apal.2024.103424","url":null,"abstract":"<div><p>We determine sufficient structure for an elementary topos to emulate Nelson's Internal Set Theory in its internal language, and show that any topos satisfying the internal axiom of choice occurs as a universe of standard objects and maps. This development allows one to employ the proof methods of nonstandard analysis (transfer, standardisation, and idealisation) in new environments such as toposes of <em>G</em>-sets and Boolean étendues.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"175 5","pages":"Article 103424"},"PeriodicalIF":0.8,"publicationDate":"2024-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139946240","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":"Arboreal categories and equi-resource homomorphism preservation theorems","authors":"Samson Abramsky,&nbsp;Luca Reggio","doi":"10.1016/j.apal.2024.103423","DOIUrl":"10.1016/j.apal.2024.103423","url":null,"abstract":"<div><p>The classical homomorphism preservation theorem, due to Łoś, Lyndon and Tarski, states that a first-order sentence <em>φ</em> is preserved under homomorphisms between structures if, and only if, it is equivalent to an existential positive sentence <em>ψ</em>. Given a notion of (syntactic) complexity of sentences, an “equi-resource” homomorphism preservation theorem improves on the classical result by ensuring that <em>ψ</em> can be chosen so that its complexity does not exceed that of <em>φ</em>.</p><p>We describe an axiomatic approach to equi-resource homomorphism preservation theorems based on the notion of arboreal category. This framework is then employed to establish novel homomorphism preservation results, and improve on known ones, for various logic fragments, including first-order, guarded and modal logics.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"175 6","pages":"Article 103423"},"PeriodicalIF":0.8,"publicationDate":"2024-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0168007224000204/pdfft?md5=483bf3d114b061fe423ab82a314823cd&pid=1-s2.0-S0168007224000204-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139925039","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":"A new model construction by making a detour via intuitionistic theories IV: A closer connection between KPω and BI","authors":"Kentaro Sato","doi":"10.1016/j.apal.2024.103422","DOIUrl":"10.1016/j.apal.2024.103422","url":null,"abstract":"<div><p>By combining tree representation of sets with the method introduced in the previous three papers I–III <span>[39]</span>, <span>[35]</span>, <span>[37]</span> in the series, we give a new <span><math><msubsup><mrow><mi>Π</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>1</mn></mrow></msubsup></math></span>-preserving interpretation of <span><math><mrow><mi>KP</mi></mrow><msub><mrow><mi>ω</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>+</mo><mo>(</mo><msub><mrow><mi>Π</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow></msub><mtext>-</mtext><mrow><mi>Found</mi></mrow><mo>)</mo><mo>+</mo><mi>θ</mi></math></span> (Kripke–Platek set theory with the foundation schema restricted to <span><math><msub><mrow><mi>Π</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow></msub></math></span>, and augmented by <em>θ</em>) in <span><math><msubsup><mrow><mi>Σ</mi></mrow><mrow><mn>1</mn></mrow><mrow><mn>1</mn></mrow></msubsup><mtext>-</mtext><msub><mrow><mi>AC</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>+</mo><mo>(</mo><msubsup><mrow><mi>Π</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow><mrow><mn>1</mn></mrow></msubsup><mtext>-</mtext><mrow><mi>TI</mi></mrow><mo>)</mo><mo>+</mo><mi>θ</mi></math></span> for any <span><math><msubsup><mrow><mi>Π</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>1</mn></mrow></msubsup></math></span> sentence <em>θ</em>, where the language of second order arithmetic is considered as a sublanguage of that of set theory via the standard interpretation. Thus the addition of any <span><math><msubsup><mrow><mi>Π</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>1</mn></mrow></msubsup></math></span> theorem of <span><math><mrow><mi>BI</mi></mrow><mo>≡</mo><msubsup><mrow><mi>Σ</mi></mrow><mrow><mn>1</mn></mrow><mrow><mn>1</mn></mrow></msubsup><mtext>-</mtext><msub><mrow><mi>AC</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>+</mo><mo>(</mo><msubsup><mrow><mi>Π</mi></mrow><mrow><mo>∞</mo></mrow><mrow><mn>1</mn></mrow></msubsup><mtext>-</mtext><mrow><mi>TI</mi></mrow><mo>)</mo></math></span> does not increase the consistency strength of <strong>KP</strong><em>ω</em>. Among such <span><math><msubsup><mrow><mi>Π</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>1</mn></mrow></msubsup></math></span> theorems are several fixed point principles for positive arithmetical operators and <em>ω</em>-model reflection (the cofinal existence of coded <em>ω</em>-models) for theorems of <strong>BI</strong>. The reader's familiarity to the previous works I–III in the series might help, but is not necessary.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"175 7","pages":"Article 103422"},"PeriodicalIF":0.8,"publicationDate":"2024-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139811673","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 ergodicity phenomena for Bernoulli shifts of bounded algebraic dimension","authors":"Aristotelis Panagiotopoulos ,&nbsp;Assaf Shani","doi":"10.1016/j.apal.2024.103412","DOIUrl":"10.1016/j.apal.2024.103412","url":null,"abstract":"<div><p>The algebraic dimension of a Polish permutation group <span><math><mi>Q</mi><mo>≤</mo><mrow><mi>Sym</mi></mrow><mo>(</mo><mi>N</mi><mo>)</mo></math></span> is the size of the largest <span><math><mi>A</mi><mo>⊆</mo><mi>N</mi></math></span> with the property that the orbit of every <span><math><mi>a</mi><mo>∈</mo><mi>A</mi></math></span> under the pointwise stabilizer of <span><math><mi>A</mi><mo>∖</mo><mo>{</mo><mi>a</mi><mo>}</mo></math></span> is infinite. We study the Bernoulli shift <span><math><mi>P</mi><mo>↷</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup></math></span> for various Polish permutation groups <em>P</em> and we provide criteria under which the <em>P</em>-shift is generically ergodic relative to the injective part of the <em>Q</em>-shift, when <em>Q</em> has algebraic dimension ≤<em>n</em>. We use this to show that the sequence of pairwise ⁎-reduction-incomparable equivalence relations defined in <span>[18]</span> is a strictly increasing sequence in the Borel reduction hierarchy. We also use our main theorem to exhibit an equivalence relation of pinned cardinal <span><math><msubsup><mrow><mi>ℵ</mi></mrow><mrow><mn>1</mn></mrow><mrow><mo>+</mo></mrow></msubsup></math></span> which strongly resembles the equivalence relation of pinned cardinal <span><math><msubsup><mrow><mi>ℵ</mi></mrow><mrow><mn>1</mn></mrow><mrow><mo>+</mo></mrow></msubsup></math></span> from <span>[25]</span>, but which does not Borel reduce to the latter. It remains open whether they are actually incomparable under Borel reductions.</p><p>Our proofs rely on the study of symmetric models whose symmetries come from the group <em>Q</em>. We show that when <em>Q</em> is “locally finite”—e.g. when <span><math><mi>Q</mi><mo>=</mo><mrow><mi>Aut</mi></mrow><mo>(</mo><mi>M</mi><mo>)</mo></math></span>, where <span><math><mi>M</mi></math></span> is the Fraïssé limit of a Fraïssé class satisfying the disjoint amalgamation property—the corresponding symmetric model admits a theory of supports which is analogous to that in the basic Cohen model.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"175 5","pages":"Article 103412"},"PeriodicalIF":0.8,"publicationDate":"2024-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139657003","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 formal verification of the ctm approach to forcing","authors":"Emmanuel Gunther ,&nbsp;Miguel Pagano ,&nbsp;Pedro Sánchez Terraf ,&nbsp;Matías Steinberg","doi":"10.1016/j.apal.2024.103413","DOIUrl":"10.1016/j.apal.2024.103413","url":null,"abstract":"<div><p>We discuss some highlights of our computer-verified proof of the construction, given a countable transitive set-model <em>M</em> of <em>ZFC</em>, of generic extensions satisfying <span><math><mrow><mi>ZFC</mi></mrow><mo>+</mo><mo>¬</mo><mrow><mi>CH</mi></mrow></math></span> and <span><math><mrow><mi>ZFC</mi></mrow><mo>+</mo><mrow><mi>CH</mi></mrow></math></span>. Moreover, let <span><math><mi>R</mi></math></span> be the set of instances of the Axiom of Replacement. We isolated a 21-element subset <span><math><mi>Ω</mi><mo>⊆</mo><mi>R</mi></math></span> and defined <span><math><mi>F</mi><mo>:</mo><mi>R</mi><mo>→</mo><mi>R</mi></math></span> such that for every <span><math><mi>Φ</mi><mo>⊆</mo><mi>R</mi></math></span> and <em>M</em>-generic <em>G</em>, <span><math><mi>M</mi><mo>⊨</mo><mrow><mi>ZC</mi></mrow><mo>∪</mo><mi>F</mi><mtext>“</mtext><mi>Φ</mi><mo>∪</mo><mi>Ω</mi></math></span> implies <span><math><mi>M</mi><mo>[</mo><mi>G</mi><mo>]</mo><mo>⊨</mo><mrow><mi>ZC</mi></mrow><mo>∪</mo><mi>Φ</mi><mo>∪</mo><mo>{</mo><mo>¬</mo><mrow><mi>CH</mi></mrow><mo>}</mo></math></span>, where <em>ZC</em> is Zermelo set theory with Choice.</p><p>To achieve this, we worked in the proof assistant <em>Isabelle</em>, basing our development on the Isabelle/ZF library by L. Paulson and others.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"175 5","pages":"Article 103413"},"PeriodicalIF":0.8,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139649384","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":"Sharp Vaught's conjecture for some classes of partial orders","authors":"Miloš S. Kurilić","doi":"10.1016/j.apal.2024.103411","DOIUrl":"10.1016/j.apal.2024.103411","url":null,"abstract":"<div><p>Matatyahu Rubin has shown that a sharp version of Vaught's conjecture, <span><math><mi>I</mi><mo>(</mo><mi>T</mi><mo>,</mo><mi>ω</mi><mo>)</mo><mo>∈</mo><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mi>c</mi><mo>}</mo></math></span>, holds for each complete theory of linear order <span><math><mi>T</mi></math></span>. We show that the same is true for each complete theory of partial order having a model in the minimal class of partial orders containing the class of linear orders and which is closed under finite products and finite disjoint unions. The same holds for the extension of the class of rooted trees admitting a finite monomorphic decomposition, obtained in the same way. The sharp version of Vaught's conjecture also holds for the theories of trees which are infinite disjoint unions of linear orders.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"175 4","pages":"Article 103411"},"PeriodicalIF":0.8,"publicationDate":"2024-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139462598","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":"Towards characterizing the >ω2-fickle recursively enumerable Turing degrees","authors":"Liling Ko","doi":"10.1016/j.apal.2023.103403","DOIUrl":"10.1016/j.apal.2023.103403","url":null,"abstract":"<div><p><span>Given a finite lattice </span><em>L</em><span> that can be embedded in the recursively enumerable (r.e.) Turing degrees </span><span><math><mo>〈</mo><msub><mrow><mi>R</mi></mrow><mrow><mi>T</mi></mrow></msub><mo>,</mo><msub><mrow><mo>≤</mo></mrow><mrow><mi>T</mi></mrow></msub><mo>〉</mo></math></span>, it is not known how one can characterize the degrees <span><math><mi>d</mi><mo>∈</mo><msub><mrow><mi>R</mi></mrow><mrow><mi>T</mi></mrow></msub></math></span> below which <em>L</em> can be embedded. Two important characterizations are of the <span><math><msub><mrow><mi>L</mi></mrow><mrow><mn>7</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>M</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span> lattices, where the lattices are embedded below <strong>d</strong> if and only if <strong>d</strong> contains sets of “<em>fickleness</em>” &gt;<em>ω</em> and <span><math><mo>≥</mo><msup><mrow><mi>ω</mi></mrow><mrow><mi>ω</mi></mrow></msup></math></span> respectively. We work towards finding a lattice that characterizes the levels above <span><math><msup><mrow><mi>ω</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>, the first non-trivial level after <em>ω</em>. We considered lattices that are as “short” in height and “narrow” in width as <span><math><msub><mrow><mi>L</mi></mrow><mrow><mn>7</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>M</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span>, but the lattices characterize also the &gt;<em>ω</em> or <span><math><mo>≥</mo><msup><mrow><mi>ω</mi></mrow><mrow><mi>ω</mi></mrow></msup></math></span> levels, if the lattices are not already embeddable below all non-zero r.e. degrees. We also considered upper semilattices (USLs) by removing the bottom meet(s) of some previously considered lattices, but the removals did not change the levels characterized. We discovered three lattices besides <span><math><msub><mrow><mi>M</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span> that also characterize the <span><math><mo>≥</mo><msup><mrow><mi>ω</mi></mrow><mrow><mi>ω</mi></mrow></msup></math></span>-levels. Our search for <span><math><mo>&gt;</mo><msup><mrow><mi>ω</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-candidates can therefore be reduced to the lattice-theoretic problem of finding lattices that do not contain any of the four <span><math><mo>≥</mo><msup><mrow><mi>ω</mi></mrow><mrow><mi>ω</mi></mrow></msup></math></span><span>-lattices as sublattices.</span></p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"175 4","pages":"Article 103403"},"PeriodicalIF":0.8,"publicationDate":"2024-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139104768","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":"Structural and universal completeness in algebra and logic","authors":"Paolo Aglianò ,&nbsp;Sara Ugolini","doi":"10.1016/j.apal.2023.103391","DOIUrl":"10.1016/j.apal.2023.103391","url":null,"abstract":"<div><p>In this work we study the notions of structural and universal completeness both from the algebraic and logical point of view. In particular, we provide new algebraic characterizations of quasivarieties that are actively and passively universally complete, and passively structurally complete. We apply these general results to varieties of bounded lattices and to quasivarieties related to substructural logics. In particular we show that a substructural logic satisfying weakening is passively structurally complete if and only if every classical contradiction is explosive in it. Moreover, we fully characterize the passively structurally complete varieties of <span><math><mi>MTL</mi></math></span>-algebras, i.e., bounded commutative integral residuated lattices generated by chains.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"175 3","pages":"Article 103391"},"PeriodicalIF":0.8,"publicationDate":"2023-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0168007223001483/pdfft?md5=566d39a3ec46ed86a39bb7490723fd1b&pid=1-s2.0-S0168007223001483-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138683789","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}