The Journal of Symbolic Logic最新文献

{"title":"ON RANK NOT ONLY IN NSOP THEORIES","authors":"JAN DOBROWOLSKI, DANIEL MAX HOFFMANN","doi":"10.1017/jsl.2024.9","DOIUrl":"https://doi.org/10.1017/jsl.2024.9","url":null,"abstract":"<p>We introduce a family of local ranks <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240327105301107-0039:S0022481224000094:S0022481224000094_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$D_Q$</span></span></img></span></span> depending on a finite set <span>Q</span> of pairs of the form <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240327105301107-0039:S0022481224000094:S0022481224000094_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$(varphi (x,y),q(y)),$</span></span></img></span></span> where <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240327105301107-0039:S0022481224000094:S0022481224000094_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$varphi (x,y)$</span></span></img></span></span> is a formula and <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240327105301107-0039:S0022481224000094:S0022481224000094_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$q(y)$</span></span></img></span></span> is a global type. We prove that in any NSOP<span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240327105301107-0039:S0022481224000094:S0022481224000094_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$_1$</span></span></img></span></span> theory these ranks satisfy some desirable properties; in particular, <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240327105301107-0039:S0022481224000094:S0022481224000094_inline7.png\"><span data-mathjax-type=\"texmath\"><span>$D_Q(x=x)&lt;omega $</span></span></img></span></span> for any finite tuple of variables <span>x</span> and any <span>Q</span>, if <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240327105301107-0039:S0022481224000094:S0022481224000094_inline8.png\"><span data-mathjax-type=\"texmath\"><span>$qsupseteq p$</span></span></img></span></span> is a Kim-forking extension of types, then <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240327105301107-0039:S0022481224000094:S0022481224000094_inline9.png\"><span data-mathjax-type=\"texmath\"><span>$D_Q(q)&lt;D_Q(p)$</span></span></img></span></span> for some <span>Q</span>, and if <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240327105301107-0039:S0022481224000094:S0022481224000094_inline10.png\"><span data-mathjax-type=\"texmath\"><span>$qsupseteq p$</span></spa","PeriodicalId":501300,"journal":{"name":"The Journal of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140587758","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A HIERARCHY ON NON-ARCHIMEDEAN POLISH GROUPS ADMITTING A COMPATIBLE COMPLETE LEFT-INVARIANT METRIC","authors":"LONGYUN DING, XU WANG","doi":"10.1017/jsl.2024.7","DOIUrl":"https://doi.org/10.1017/jsl.2024.7","url":null,"abstract":"<p>In this article, we introduce a hierarchy on the class of non-archimedean Polish groups that admit a compatible complete left-invariant metric. We denote this hierarchy by <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240226125007514-0272:S0022481224000070:S0022481224000070_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$alpha $</span></span></img></span></span>-CLI and L-<span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240226125007514-0272:S0022481224000070:S0022481224000070_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$alpha $</span></span></img></span></span>-CLI where <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240226125007514-0272:S0022481224000070:S0022481224000070_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$alpha $</span></span></img></span></span> is a countable ordinal. We establish three results: </p><ol><li><p><span>(1)</span> <span>G</span> is <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240226125007514-0272:S0022481224000070:S0022481224000070_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$0$</span></span></img></span></span>-CLI iff <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240226125007514-0272:S0022481224000070:S0022481224000070_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$G={1_G}$</span></span></img></span></span>;</p></li><li><p><span>(2)</span> <span>G</span> is <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240226125007514-0272:S0022481224000070:S0022481224000070_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$1$</span></span></img></span></span>-CLI iff <span>G</span> admits a compatible complete two-sided invariant metric; and</p></li><li><p><span>(3)</span> <span>G</span> is L-<span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240226125007514-0272:S0022481224000070:S0022481224000070_inline7.png\"><span data-mathjax-type=\"texmath\"><span>$alpha $</span></span></img></span></span>-CLI iff <span>G</span> is locally <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240226125007514-0272:S0022481224000070:S0022481224000070_inline8.png\"><span data-mathjax-type=\"texmath\"><span>$alpha $</span></span></img></span></span>-CLI, i.e., <span>G</span> contains an open subgroup that is <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.","PeriodicalId":501300,"journal":{"name":"The Journal of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140011282","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"BETWEENNESS ALGEBRAS","authors":"IVO DÜNTSCH, RAFAŁ GRUSZCZYŃSKI, PAULA MENCHÓN","doi":"10.1017/jsl.2023.86","DOIUrl":"https://doi.org/10.1017/jsl.2023.86","url":null,"abstract":"We introduce and study a class of <jats:italic>betweenness algebras</jats:italic>—Boolean algebras with binary operators, closely related to ternary frames with a betweenness relation. From various axioms for betweenness, we chose those that are most common, which makes our work applicable to a wide range of betweenness structures studied in the literature. On the algebraic side, we work with two operators of <jats:italic>possibility</jats:italic> and of <jats:italic>sufficiency</jats:italic>.","PeriodicalId":501300,"journal":{"name":"The Journal of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139760278","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"THE PENTAGON AS A SUBSTRUCTURE LATTICE OF MODELS OF PEANO ARITHMETIC","authors":"JAMES H. SCHMERL","doi":"10.1017/jsl.2024.6","DOIUrl":"https://doi.org/10.1017/jsl.2024.6","url":null,"abstract":"<p>Wilkie proved in 1977 that every countable model <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240426094705323-0134:S0022481224000069:S0022481224000069_inline1.png\"><span data-mathjax-type=\"texmath\"><span>${mathcal M}$</span></span></img></span></span> of Peano Arithmetic has an elementary end extension <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240426094705323-0134:S0022481224000069:S0022481224000069_inline2.png\"><span data-mathjax-type=\"texmath\"><span>${mathcal N}$</span></span></img></span></span> such that the interstructure lattice <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240426094705323-0134:S0022481224000069:S0022481224000069_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$operatorname {mathrm {Lt}}({mathcal N} / {mathcal M})$</span></span></img></span></span> is the pentagon lattice <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240426094705323-0134:S0022481224000069:S0022481224000069_inline4.png\"><span data-mathjax-type=\"texmath\"><span>${mathbf N}_5$</span></span></img></span></span>. This theorem implies that every countable nonstandard <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240426094705323-0134:S0022481224000069:S0022481224000069_inline5.png\"><span data-mathjax-type=\"texmath\"><span>${mathcal M}$</span></span></img></span></span> has an elementary cofinal extension <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240426094705323-0134:S0022481224000069:S0022481224000069_inline6.png\"><span data-mathjax-type=\"texmath\"><span>${mathcal N}$</span></span></img></span></span> such that <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240426094705323-0134:S0022481224000069:S0022481224000069_inline7.png\"><span data-mathjax-type=\"texmath\"><span>$operatorname {mathrm {Lt}}({mathcal N} / {mathcal M}) cong {mathbf N}_5$</span></span></img></span></span>. It is proved here that whenever <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240426094705323-0134:S0022481224000069:S0022481224000069_inline8.png\"><span data-mathjax-type=\"texmath\"><span>${mathcal M} prec {mathcal N} models mathsf {PA}$</span></span></img></span></span> and <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240426094705323-0134:S0022481224000069:S0022481224000069_inline9.png\"><span dat","PeriodicalId":501300,"journal":{"name":"The Journal of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140811420","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"REGAININGLY APPROXIMABLE NUMBERS AND SETS","authors":"PETER HERTLING, RUPERT HÖLZL, PHILIP JANICKI","doi":"10.1017/jsl.2024.5","DOIUrl":"https://doi.org/10.1017/jsl.2024.5","url":null,"abstract":"<p>We call an <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240227060042912-0720:S0022481224000057:S0022481224000057_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$alpha in mathbb {R}$</span></span></img></span></span> <span>regainingly approximable</span> if there exists a computable nondecreasing sequence <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240227060042912-0720:S0022481224000057:S0022481224000057_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$(a_n)_n$</span></span></img></span></span> of rational numbers converging to <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240227060042912-0720:S0022481224000057:S0022481224000057_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$alpha $</span></span></img></span></span> with <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240227060042912-0720:S0022481224000057:S0022481224000057_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$alpha - a_n &lt; 2^{-n}$</span></span></img></span></span> for infinitely many <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240227060042912-0720:S0022481224000057:S0022481224000057_inline5.png\"><span data-mathjax-type=\"texmath\"><span>${n in mathbb {N}}$</span></span></img></span></span>. We also call a set <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240227060042912-0720:S0022481224000057:S0022481224000057_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$Asubseteq mathbb {N}$</span></span></img></span></span> <span>regainingly approximable</span> if it is c.e. and the strongly left-computable number <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240227060042912-0720:S0022481224000057:S0022481224000057_inline7.png\"><span data-mathjax-type=\"texmath\"><span>$2^{-A}$</span></span></img></span></span> is regainingly approximable. We show that the set of regainingly approximable sets is neither closed under union nor intersection and that every c.e. Turing degree contains such a set. Furthermore, the regainingly approximable numbers lie properly between the computable and the left-computable numbers and are not closed under addition. While regainingly approximable numbers are easily seen to be i.o. <span>K</span>-trivial, we construct such an <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240227060042912-0720:S0022481224000057:S0022481224000057_inline8.png\"","PeriodicalId":501300,"journal":{"name":"The Journal of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140007945","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"POLISH SPACE PARTITION PRINCIPLES AND THE HALPERN-LÄUCHLI THEOREM","authors":"C. Lambie-Hanson, Andy Zucker","doi":"10.1017/jsl.2024.4","DOIUrl":"https://doi.org/10.1017/jsl.2024.4","url":null,"abstract":"","PeriodicalId":501300,"journal":{"name":"The Journal of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139612799","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"DEGREE OF SATISFIABILITY IN HEYTING ALGEBRAS","authors":"BENJAMIN MERLIN BUMPUS, ZOLTAN A. KOCSIS","doi":"10.1017/jsl.2024.2","DOIUrl":"https://doi.org/10.1017/jsl.2024.2","url":null,"abstract":"<p>We investigate degree of satisfiability questions in the context of Heyting algebras and intuitionistic logic. We classify all equations in one free variable with respect to finite satisfiability gap, and determine which common principles of classical logic in multiple free variables have finite satisfiability gap. In particular we prove that, in a finite non-Boolean Heyting algebra, the probability that a randomly chosen element satisfies <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240226124105987-0144:S0022481224000021:S0022481224000021_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$x vee neg x = top $</span></span></img></span></span> is no larger than <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240226124105987-0144:S0022481224000021:S0022481224000021_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$frac {2}{3}$</span></span></img></span></span>. Finally, we generalize our results to infinite Heyting algebras, and present their applications to point-set topology, black-box algebras, and the philosophy of logic.</p>","PeriodicalId":501300,"journal":{"name":"The Journal of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-01-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140007920","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"THE BAIRE CLOSURE AND ITS LOGIC","authors":"G. BEZHANISHVILI, D. FERNÁNDEZ-DUQUE","doi":"10.1017/jsl.2024.1","DOIUrl":"https://doi.org/10.1017/jsl.2024.1","url":null,"abstract":"<p>The Baire algebra of a topological space <span>X</span> is the quotient of the algebra of all subsets of <span>X</span> modulo the meager sets. We show that this Boolean algebra can be endowed with a natural closure operator, resulting in a closure algebra which we denote <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240127131651504-0594:S002248122400001X:S002248122400001X_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$mathbf {Baire}(X)$</span></span></img></span></span>. We identify the modal logic of such algebras to be the well-known system <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240127131651504-0594:S002248122400001X:S002248122400001X_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$mathsf {S5}$</span></span></img></span></span>, and prove soundness and strong completeness for the cases where <span>X</span> is crowded and either completely metrizable and continuum-sized or locally compact Hausdorff. We also show that every extension of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240127131651504-0594:S002248122400001X:S002248122400001X_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$mathsf {S5}$</span></span></img></span></span> is the modal logic of a subalgebra of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240127131651504-0594:S002248122400001X:S002248122400001X_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$mathbf {Baire}(X)$</span></span></img></span></span>, and that soundness and strong completeness also holds in the language with the universal modality.</p>","PeriodicalId":501300,"journal":{"name":"The Journal of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139582440","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Note on implying","authors":"Sean Cody","doi":"10.1017/jsl.2023.98","DOIUrl":"https://doi.org/10.1017/jsl.2023.98","url":null,"abstract":"","PeriodicalId":501300,"journal":{"name":"The Journal of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139385176","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"ON COMPACTNESS OF WEAK SQUARE AT SINGULARS OF UNCOUNTABLE COFINALITY","authors":"MAXWELL LEVINE","doi":"10.1017/jsl.2023.101","DOIUrl":"https://doi.org/10.1017/jsl.2023.101","url":null,"abstract":"<p>Cummings, Foreman, and Magidor proved that Jensen’s square principle is non-compact at <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240418113432529-0233:S0022481223001019:S0022481223001019_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$aleph _omega $</span></span></img></span></span>, meaning that it is consistent that <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240418113432529-0233:S0022481223001019:S0022481223001019_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$square _{aleph _n}$</span></span></img></span></span> holds for all <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240418113432529-0233:S0022481223001019:S0022481223001019_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$n&lt;omega $</span></span></img></span></span> while <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240418113432529-0233:S0022481223001019:S0022481223001019_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$square _{aleph _omega }$</span></span></img></span></span> fails. We investigate the natural question of whether this phenomenon generalizes to singulars of uncountable cofinality. Surprisingly, we show that under some mild <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240418113432529-0233:S0022481223001019:S0022481223001019_inline5.png\"><span data-mathjax-type=\"texmath\"><span>${{mathsf {PCF}}}$</span></span></img></span></span>-theoretic hypotheses, the weak square principle <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240418113432529-0233:S0022481223001019:S0022481223001019_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$square _kappa ^*$</span></span></img></span></span> is in fact compact at singulars of uncountable cofinality.</p>","PeriodicalId":501300,"journal":{"name":"The Journal of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140626115","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}