The Journal of Symbolic Logic最新文献

{"title":"CANTOR’S THEOREM MAY FAIL FOR FINITARY PARTITIONS","authors":"GUOZHEN SHEN","doi":"10.1017/jsl.2024.24","DOIUrl":"https://doi.org/10.1017/jsl.2024.24","url":null,"abstract":"<p>A partition is finitary if all its members are finite. For a set <span>A</span>, <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240420072412745-0886:S0022481224000240:S0022481224000240_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$mathscr {B}(A)$</span></span></img></span></span> denotes the set of all finitary partitions of <span>A</span>. It is shown consistent with <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240420072412745-0886:S0022481224000240:S0022481224000240_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$mathsf {ZF}$</span></span></img></span></span> (without the axiom of choice) that there exist an infinite set <span>A</span> and a surjection from <span>A</span> onto <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240420072412745-0886:S0022481224000240:S0022481224000240_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$mathscr {B}(A)$</span></span></img></span></span>. On the other hand, we prove in <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240420072412745-0886:S0022481224000240:S0022481224000240_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$mathsf {ZF}$</span></span></img></span></span> some theorems concerning <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240420072412745-0886:S0022481224000240:S0022481224000240_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$mathscr {B}(A)$</span></span></img></span></span> for infinite sets <span>A</span>, among which are the following: </p><ol><li><p><span>(1)</span> If there is a finitary partition of <span>A</span> without singleton blocks, then there are no surjections from <span>A</span> onto <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240420072412745-0886:S0022481224000240:S0022481224000240_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$mathscr {B}(A)$</span></span></img></span></span> and no finite-to-one functions from <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240420072412745-0886:S0022481224000240:S0022481224000240_inline7.png\"><span data-mathjax-type=\"texmath\"><span>$mathscr {B}(A)$</span></span></img></span></span> to <span>A</span>.</p></li><li><p><span>(2)</span> For all <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240420072412745-0886:S0022481224000240:S0022481224000240_inline8.png\"><span data-mathjax-type=\"texmath\"><span>$nin omega $</","PeriodicalId":501300,"journal":{"name":"The Journal of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140635393","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":"WAND/SET THEORIES: A REALIZATION OF CONWAY’S MATHEMATICIANS’ LIBERATION MOVEMENT, WITH AN APPLICATION TO CHURCH’S SET THEORY WITH A UNIVERSAL SET","authors":"TIM BUTTON","doi":"10.1017/jsl.2024.21","DOIUrl":"https://doi.org/10.1017/jsl.2024.21","url":null,"abstract":"<p>Consider a variant of the usual story about the iterative conception of sets. As usual, at every stage, you find all the (bland) sets of objects which you found earlier. But you also find the result of tapping any earlier-found object with any magic wand (from a given stock of magic wands).</p><p>By varying the number and behaviour of the wands, we can flesh out this idea in many different ways. This paper's main Theorem is that any loosely constructive way of fleshing out this idea is synonymous with a ZF-like theory.</p><p>This Theorem has rich applications; it realizes John Conway's (1976) Mathematicians' Liberation Movement; and it connects with a lovely idea due to Alonzo Church (1974).</p>","PeriodicalId":501300,"journal":{"name":"The Journal of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140811497","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":"EXTENSIONS AND LIMITS OF THE SPECKER–BLATTER THEOREM","authors":"ELDAR FISCHER, JOHANN A. MAKOWSKY","doi":"10.1017/jsl.2024.17","DOIUrl":"https://doi.org/10.1017/jsl.2024.17","url":null,"abstract":"<p>The original Specker–Blatter theorem (1983) was formulated for classes of structures <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240418114435066-0017:S0022481224000173:S0022481224000173_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$mathcal {C}$</span></span></img></span></span> of one or several binary relations definable in Monadic Second Order Logic MSOL. It states that the number of such structures on the set <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240418114435066-0017:S0022481224000173:S0022481224000173_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$[n]$</span></span></img></span></span> is modularly C-finite (MC-finite). In previous work we extended this to structures definable in CMSOL, MSOL extended with modular counting quantifiers. The first author also showed that the Specker–Blatter theorem does not hold for one quaternary relation (2003).</p><p>If the vocabulary allows a constant symbol <span>c</span>, there are <span>n</span> possible interpretations on <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240418114435066-0017:S0022481224000173:S0022481224000173_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$[n]$</span></span></img></span></span> for <span>c</span>. We say that a constant <span>c</span> is <span>hard-wired</span> if <span>c</span> is always interpreted by the same element <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240418114435066-0017:S0022481224000173:S0022481224000173_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$j in [n]$</span></span></img></span></span>. In this paper we show: </p><ol><li><p><span>(i)</span> The Specker–Blatter theorem also holds for CMSOL when hard-wired constants are allowed. The proof method of Specker and Blatter does not work in this case.</p></li><li><p><span>(ii)</span> The Specker–Blatter theorem does not hold already for <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240418114435066-0017:S0022481224000173:S0022481224000173_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$mathcal {C}$</span></span></img></span></span> with one ternary relation definable in First Order Logic FOL. This was left open since 1983.</p></li></ol><p></p><p>Using hard-wired constants allows us to show MC-finiteness of counting functions of various restricted partition functions which were not known to be MC-finite till now. Among them we have the restricted Bell numbers <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240418114435066-0017:S0022481224000","PeriodicalId":501300,"journal":{"name":"The Journal of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140625839","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 THE FRAGILITY OF INTERPOLATION","authors":"Andrzej Tarlecki","doi":"10.1017/jsl.2024.19","DOIUrl":"https://doi.org/10.1017/jsl.2024.19","url":null,"abstract":"","PeriodicalId":501300,"journal":{"name":"The Journal of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140221888","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":"THREE SURPRISING INSTANCES OF DIVIDING","authors":"GABRIEL CONANT, ALEX KRUCKMAN","doi":"10.1017/jsl.2024.20","DOIUrl":"https://doi.org/10.1017/jsl.2024.20","url":null,"abstract":"<p>We give three counterexamples to the folklore claim that in an arbitrary theory, if a complete type <span>p</span> over a set <span>B</span> does not divide over <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240412093917073-0582:S0022481224000203:S0022481224000203_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$Csubseteq B$</span></span></img></span></span>, then no extension of <span>p</span> to a complete type over <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240412093917073-0582:S0022481224000203:S0022481224000203_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$operatorname {acl}(B)$</span></span></img></span></span> divides over <span>C</span>. Two of our examples are also the first known theories where all sets are extension bases for nonforking, but forking and dividing differ for complete types (answering a question of Adler). One example is an <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240412093917073-0582:S0022481224000203:S0022481224000203_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$mathrm {NSOP}_1$</span></span></img></span></span> theory with a complete type that forks, but does not divide, over a model (answering a question of d’Elbée). Moreover, dividing independence fails to imply M-independence in this example (which refutes another folklore claim). In addition to these counterexamples, we summarize various related properties of dividing that are still true. We also address consequences for previous literature, including an earlier unpublished result about forking and dividing in free amalgamation theories, and some claims about dividing in the theory of generic <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240412093917073-0582:S0022481224000203:S0022481224000203_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$K_{m,n}$</span></span></img></span></span>-free incidence structures.</p>","PeriodicalId":501300,"journal":{"name":"The Journal of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140587757","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 LOPEZ-ESCOBAR THEOREM FOR CONTINUOUS DOMAINS","authors":"Nikolay Bazhenov, Ekaterina Fokina, D. Rossegger, Alexandra Soskova, Stefan V. Vatev","doi":"10.1017/jsl.2024.18","DOIUrl":"https://doi.org/10.1017/jsl.2024.18","url":null,"abstract":"","PeriodicalId":501300,"journal":{"name":"The Journal of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140239977","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":"TWO-CARDINAL DERIVED TOPOLOGIES, INDESCRIBABILITY AND RAMSEYNESS","authors":"BRENT CODY, CHRIS LAMBIE-HANSON, JING ZHANG","doi":"10.1017/jsl.2024.16","DOIUrl":"https://doi.org/10.1017/jsl.2024.16","url":null,"abstract":"<p>We introduce a natural two-cardinal version of Bagaria’s sequence of derived topologies on ordinals. We prove that for our sequence of two-cardinal derived topologies, limit points of sets can be characterized in terms of a new iterated form of pairwise simultaneous reflection of certain kinds of stationary sets, the first few instances of which are often equivalent to notions related to strong stationarity, which has been studied previously in the context of strongly normal ideals. The non-discreteness of these two-cardinal derived topologies can be obtained from certain two-cardinal indescribability hypotheses, which follow from local instances of supercompactness. Additionally, we answer several questions posed by the first author, Holy and White on the relationship between Ramseyness and indescribability in both the cardinal context and in the two-cardinal context.</p>","PeriodicalId":501300,"journal":{"name":"The Journal of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140587848","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 NEW PERSPECTIVE ON SEMI-RETRACTIONS AND THE RAMSEY PROPERTY","authors":"DANA BARTOŠOVÁ, LYNN SCOW","doi":"10.1017/jsl.2024.15","DOIUrl":"https://doi.org/10.1017/jsl.2024.15","url":null,"abstract":"<p>We investigate the notion of a semi-retraction between two first-order structures (in typically different signatures) that was introduced by the second author as a link between the Ramsey property and generalized indiscernible sequences. We look at semi-retractions through a new lens establishing transfers of the Ramsey property and finite Ramsey degrees under quite general conditions that are optimal as demonstrated by counterexamples. Finally, we compare semi-retractions to the category theoretic notion of a pre-adjunction.</p>","PeriodicalId":501300,"journal":{"name":"The Journal of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140635125","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":"DUALITY FOR COALGEBRAS FOR VIETORIS AND MONADICITY","authors":"MARCO ABBADINI, IVAN DI LIBERTI","doi":"10.1017/jsl.2024.14","DOIUrl":"https://doi.org/10.1017/jsl.2024.14","url":null,"abstract":"<p>We prove that the opposite of the category of coalgebras for the Vietoris endofunctor on the category of compact Hausdorff spaces is monadic over <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240326144936502-0975:S0022481224000148:S0022481224000148_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$mathsf {Set}$</span></span></img></span></span>. We deliver an analogous result for the upper, lower, and convex Vietoris endofunctors acting on the category of stably compact spaces. We provide axiomatizations of the associated (infinitary) varieties. This can be seen as a version of Jónsson–Tarski duality for modal algebras beyond the zero-dimensional setting.</p>","PeriodicalId":501300,"journal":{"name":"The Journal of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140312657","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 UNIFIED APPROACH TO HINDMAN, RAMSEY, AND VAN DER WAERDEN SPACES","authors":"RAFAŁ FILIPÓW, KRZYSZTOF KOWITZ, ADAM KWELA","doi":"10.1017/jsl.2024.8","DOIUrl":"https://doi.org/10.1017/jsl.2024.8","url":null,"abstract":"<p>For many years, there have been conducting research (e.g., by Bergelson, Furstenberg, Kojman, Kubiś, Shelah, Szeptycki, Weiss) into sequentially compact spaces that are, in a sense, topological counterparts of some combinatorial theorems, for instance, Ramsey’s theorem for coloring graphs, Hindman’s finite sums theorem, and van der Waerden’s arithmetical progressions theorem. These spaces are defined with the aid of different kinds of convergences: IP-convergence, R-convergence, and ordinary convergence.</p><p>The first aim of this paper is to present a unified approach to these various types of convergences and spaces. Then, using this unified approach, we prove some general theorems about existence of the considered spaces and show that all results obtained so far in this subject can be derived from our theorems.</p><p>The second aim of this paper is to obtain new results about the specific types of these spaces. For instance, we construct a Hausdorff Hindman space that is not an <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240229152906378-0015:S0022481224000082:S0022481224000082_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$mathcal {I}_{1/n}$</span></span></img></span></span>-space and a Hausdorff differentially compact space that is not Hindman. Moreover, we compare Ramsey spaces with other types of spaces. For instance, we construct a Ramsey space that is not Hindman and a Hindman space that is not Ramsey.</p><p>The last aim of this paper is to provide a characterization that shows when there exists a space of one considered type that is not of the other kind. This characterization is expressed in purely combinatorial manner with the aid of the so-called Katětov order that has been extensively examined for many years so far.</p><p>This paper may interest the general audience of mathematicians as the results we obtain are on the intersection of topology, combinatorics, set theory, and number theory.</p>","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":"140008231","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}