Annals of Pure and Applied Logic最新文献

{"title":"Some properties of precompletely and positively numbered sets","authors":"Marat Faizrahmanov","doi":"10.1016/j.apal.2024.103523","DOIUrl":"10.1016/j.apal.2024.103523","url":null,"abstract":"<div><div>In this paper, we prove a joint generalization of Arslanov's completeness criterion and Visser's ADN theorem for precomplete numberings which, for the Gödel numbering <span><math><mi>x</mi><mo>↦</mo><msub><mrow><mi>W</mi></mrow><mrow><mi>x</mi></mrow></msub></math></span>, has been proved by Terwijn (2018). The question of whether this joint generalization takes place in each precomplete numbering has been raised in his joint paper with Barendregt in 2019. Then we consider the properties of completeness and precompleteness of numberings in the context of the positivity property. We show that no completion of a positive numbering is a minimal cover of that numbering, and that the Turing completeness of any set <em>A</em> is equivalent to the existence of a positive precomplete <em>A</em>-computable numbering of any infinite family with positive <em>A</em>-computable numbering. In addition, we prove that each <span><math><msubsup><mrow><mi>Σ</mi></mrow><mrow><mi>n</mi></mrow><mrow><mn>0</mn></mrow></msubsup></math></span>-computable numbering (<span><math><mi>n</mi><mo>⩾</mo><mn>2</mn></math></span>) of a <span><math><msubsup><mrow><mi>Σ</mi></mrow><mrow><mi>n</mi></mrow><mrow><mn>0</mn></mrow></msubsup></math></span>-computable non-principal family has a <span><math><msubsup><mrow><mi>Σ</mi></mrow><mrow><mi>n</mi></mrow><mrow><mn>0</mn></mrow></msubsup></math></span>-computable minimal cover <em>ν</em> such that for every computable function <em>f</em> there exists an integer <em>n</em> with <span><math><mi>ν</mi><mo>(</mo><mi>f</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>)</mo><mo>=</mo><mi>ν</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span>.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 2","pages":"Article 103523"},"PeriodicalIF":0.6,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142420439","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 reducibilities and set theory","authors":"Noah Schweber","doi":"10.1016/j.apal.2024.103522","DOIUrl":"10.1016/j.apal.2024.103522","url":null,"abstract":"<div><div>We study Medvedev reducibility in the context of set theory — specifically, forcing and large cardinal hypotheses. Answering a question of Hamkins and Li <span><span>[6]</span></span>, we show that the Medvedev degrees of countable ordinals are far from linearly ordered in multiple ways, our main result here being that there is a club of ordinals which is an antichain with respect to Medvedev reducibility. We then generalize these results to arbitrary “reasonably-definable” reducibilities, under appropriate set-theoretic hypotheses.</div><div>We then turn from ordinals to general structures. We show that some of the results above yield characterizations of counterexamples to Vaught's conjecture; another applies to all situations, assigning an ordinal to any reasonable class of structures and “measure” on that class. We end by discussing some directions for future research.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 2","pages":"Article 103522"},"PeriodicalIF":0.6,"publicationDate":"2024-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142535830","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":"Dividing and forking in random hypergraphs","authors":"Hirotaka Kikyo ,&nbsp;Akito Tsuboi","doi":"10.1016/j.apal.2024.103521","DOIUrl":"10.1016/j.apal.2024.103521","url":null,"abstract":"<div><div>We investigate the class of <em>m</em>-hypergraphs in which substructures with <em>l</em> elements have more than <em>s</em> subsets of size <em>m</em> that do not form a hyperedge. The class has a (unique) Fraïssé limit, if <span><math><mn>0</mn><mo>≤</mo><mi>s</mi><mo>&lt;</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>l</mi><mo>−</mo><mn>2</mn></mrow></mtd></mtr><mtr><mtd><mrow><mi>m</mi><mo>−</mo><mn>2</mn></mrow></mtd></mtr></mtable><mo>)</mo></mrow></math></span>. We show that the theory of the Fraïssé limit has <em>SU</em>-rank one if <span><math><mn>0</mn><mo>≤</mo><mi>s</mi><mo>&lt;</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>l</mi><mo>−</mo><mn>3</mn></mrow></mtd></mtr><mtr><mtd><mrow><mi>m</mi><mo>−</mo><mn>3</mn></mrow></mtd></mtr></mtable><mo>)</mo></mrow></math></span>, and dividing and forking will be different concepts in the theory if <span><math><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>l</mi><mo>−</mo><mn>3</mn></mrow></mtd></mtr><mtr><mtd><mrow><mi>m</mi><mo>−</mo><mn>3</mn></mrow></mtd></mtr></mtable><mo>)</mo></mrow><mo>≤</mo><mi>s</mi><mo>&lt;</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>l</mi><mo>−</mo><mn>2</mn></mrow></mtd></mtr><mtr><mtd><mrow><mi>m</mi><mo>−</mo><mn>2</mn></mrow></mtd></mtr></mtable><mo>)</mo></mrow></math></span>.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 2","pages":"Article 103521"},"PeriodicalIF":0.6,"publicationDate":"2024-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142326876","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":"Saturation properties for compositional truth with propositional correctness","authors":"Bartosz Wcisło","doi":"10.1016/j.apal.2024.103512","DOIUrl":"10.1016/j.apal.2024.103512","url":null,"abstract":"<div><p>It is an open question whether compositional truth with the principle of propositional soundness: “All arithmetical sentences which are propositional tautologies are true” is conservative over Peano Arithmetic. In this article, we show that the principle of propositional soundness imposes some saturation-like properties on the truth predicate, thus showing significant limitations to the possible conservativity proof.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 2","pages":"Article 103512"},"PeriodicalIF":0.6,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0168007224001167/pdfft?md5=93f2e704b024dfc73e7a30a7ab95c178&pid=1-s2.0-S0168007224001167-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142168552","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":"Foundations of iterated star maps and their use in combinatorics","authors":"Mauro Di Nasso ,&nbsp;Renling Jin","doi":"10.1016/j.apal.2024.103511","DOIUrl":"10.1016/j.apal.2024.103511","url":null,"abstract":"<div><p>We develop a framework for nonstandard analysis that gives foundations to the interplay between external and internal iterations of the star map, and we present a few examples to show the strength and flexibility of such a nonstandard technique for applications in combinatorial number theory.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 1","pages":"Article 103511"},"PeriodicalIF":0.6,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142136431","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":"Theories of Frege structure equivalent to Feferman's system T0","authors":"Daichi Hayashi","doi":"10.1016/j.apal.2024.103510","DOIUrl":"10.1016/j.apal.2024.103510","url":null,"abstract":"<div><p>Feferman <span><span>[9]</span></span> defines an impredicative system <span><math><msub><mrow><mi>T</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> of explicit mathematics, which is proof-theoretically equivalent to the subsystem <figure><img></figure> of second-order arithmetic. In this paper, we propose several systems of Frege structure with the same proof-theoretic strength as <span><math><msub><mrow><mi>T</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>. To be precise, we first consider the Kripke–Feferman theory, which is one of the most famous truth theories, and we extend it by two kinds of induction principles inspired by <span><span>[22]</span></span>. In addition, we give similar results for the system based on Aczel's original Frege structure <span><span>[1]</span></span>. Finally, we equip Cantini's supervaluation-style theory with the notion of universes, the strength of which was an open problem in <span><span>[24]</span></span>.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 1","pages":"Article 103510"},"PeriodicalIF":0.6,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142099250","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":"Universal proof theory: Semi-analytic rules and Craig interpolation","authors":"Amirhossein Akbar Tabatabai ,&nbsp;Raheleh Jalali","doi":"10.1016/j.apal.2024.103509","DOIUrl":"10.1016/j.apal.2024.103509","url":null,"abstract":"<div><p>We provide a general and syntactically defined family of sequent calculi, called <em>semi-analytic</em>, to formalize the informal notion of a “nice” sequent calculus. We show that any sufficiently strong (multimodal) substructural logic with a semi-analytic sequent calculus enjoys the Craig Interpolation Property, CIP. As a positive application, our theorem provides a uniform and modular method to prove the CIP for several multimodal substructural logics, including many fragments and variants of linear logic. More interestingly, on the negative side, it employs the lack of the CIP in almost all substructural, superintuitionistic and modal logics to provide a formal proof for the well-known intuition that almost all logics do not have a “nice” sequent calculus. More precisely, we show that many substructural logics including <span><math><mi>U</mi><msup><mrow><mi>L</mi></mrow><mrow><mo>−</mo></mrow></msup></math></span>, <span><math><mi>MTL</mi></math></span>, <span><math><mi>R</mi></math></span>, <span>Ł</span>_<em>n</em> (for <span><math><mi>n</mi><mo>⩾</mo><mn>3</mn></math></span>), <span>G</span>_<em>n</em> (for <span><math><mi>n</mi><mo>⩾</mo><mn>4</mn></math></span>), and almost all extensions of <span><math><mi>IMTL</mi></math></span>, <span><math><mi>Ł</mi></math></span>, <span><math><mi>BL</mi></math></span>, <span><math><mi>R</mi><msup><mrow><mi>M</mi></mrow><mrow><mi>e</mi></mrow></msup></math></span>, <span><math><mi>IPC</mi></math></span>, <span><math><mi>S4</mi></math></span>, and <span><math><mi>Grz</mi></math></span> (except for at most 1, 1, 3, 8, 7, 37, and 6 of them, respectively) do not have a semi-analytic calculus.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 1","pages":"Article 103509"},"PeriodicalIF":0.6,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142087766","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":"Unification types and union splittings in intermediate logics","authors":"Wojciech Dzik ,&nbsp;Sławomir Kost ,&nbsp;Piotr Wojtylak","doi":"10.1016/j.apal.2024.103508","DOIUrl":"10.1016/j.apal.2024.103508","url":null,"abstract":"<div><p>We classify intermediate logics according to their unification types. There are exactly two minimal intermediate logics with hereditary finitary unification: the least logic with hereditary unitary unification and the least logic with hereditary projective proximity (a notion close to projective approximation of Ghilardi <span><span>[17]</span></span>, <span><span>[18]</span></span>), see Figure 4. They are locally tabular and are union splittings in the lattice <span>Ext INT</span>. There are exactly four maximal intermediate logics with nullary unification (see Figure 21) and they are tabular. Any intermediate logic with neither hereditary unitary unification nor with hereditary projective proximity is included in one of the four logics. There are logics with finitary/unitary (but not hereditary finitary) unification scattered among the majority of those with nullary unification, see Figure 23. Our main tools are the characterization of locally tabular logics with finitary (or unitary) unification, by their Kripke models <span><span>[12]</span></span>, <span><span>[13]</span></span> and splittings.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 1","pages":"Article 103508"},"PeriodicalIF":0.6,"publicationDate":"2024-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142121774","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 logical and computational properties of the Vitali covering theorem","authors":"Dag Normann ,&nbsp;Sam Sanders","doi":"10.1016/j.apal.2024.103505","DOIUrl":"10.1016/j.apal.2024.103505","url":null,"abstract":"<div><p>We study a version of the Vitali covering theorem, which we call <span><math><mtext>WHBU</mtext></math></span> and which is a direct weakening of the Heine-Borel theorem for uncountable coverings, called <span><math><mtext>HBU</mtext></math></span>. We show that <span><math><mtext>WHBU</mtext></math></span> is central to measure theory by deriving it from various central approximation results related to <em>Littlewood's three principles</em>. A natural question is then <em>how hard</em> it is to prove <span><math><mtext>WHBU</mtext></math></span> (in the sense of Kohlenbach's <em>higher-order Reverse Mathematics</em>), and <em>how hard</em> it is to compute the objects claimed to exist by <span><math><mtext>WHBU</mtext></math></span> (in the sense of Kleene's computation schemes S1-S9). The answer to both questions is ‘extremely hard’, as follows: on one hand, in terms of the usual scale of (conventional) comprehension axioms, <span><math><mtext>WHBU</mtext></math></span> is only provable using Kleene's <span><math><msup><mrow><mo>∃</mo></mrow><mrow><mn>3</mn></mrow></msup></math></span>, which implies full second-order arithmetic. On the other hand, realisers (aka witnessing functionals) for <span><math><mtext>WHBU</mtext></math></span>, so-called Λ-functionals, are computable from Kleene's <span><math><msup><mrow><mo>∃</mo></mrow><mrow><mn>3</mn></mrow></msup></math></span>, but not from weaker comprehension functionals. Despite this hardness, we show that <span><math><mtext>WHBU</mtext></math></span>, and certain Λ-functionals, behave much better than <span><math><mtext>HBU</mtext></math></span> and the associated class of realisers, called Θ-functionals. In particular, we identify a specific Λ-functional called <span><math><msub><mrow><mi>Λ</mi></mrow><mrow><mtext>S</mtext></mrow></msub></math></span> which adds no computational power to the <em>Suslin functional</em>, in contrast to Θ-functionals. Finally, we introduce a hierarchy involving Θ-functionals and <span><math><mtext>HBU</mtext></math></span>.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 1","pages":"Article 103505"},"PeriodicalIF":0.6,"publicationDate":"2024-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S016800722400109X/pdfft?md5=b0cd166fc40894dfc35586ee4d3fca4b&pid=1-s2.0-S016800722400109X-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142049978","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":"Groups definable in Presburger arithmetic","authors":"Juan Pablo Acosta","doi":"10.1016/j.apal.2024.103507","DOIUrl":"10.1016/j.apal.2024.103507","url":null,"abstract":"<div><p>Here we give a complete list of the groups definable in Presburger arithmetic up to a finite index subgroup.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 1","pages":"Article 103507"},"PeriodicalIF":0.6,"publicationDate":"2024-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0168007224001118/pdfft?md5=6b2c5fc7ad959b197406d9b1a92b6a8b&pid=1-s2.0-S0168007224001118-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141978287","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}