Archive for Mathematical Logic最新文献

{"title":"The externally definable Ramsey property and fixed points on type spaces","authors":"Nadav Meir,&nbsp;Rob Sullivan","doi":"10.1007/s00153-024-00950-5","DOIUrl":"10.1007/s00153-024-00950-5","url":null,"abstract":"<div><p>We discuss the externally definable Ramsey property, a weakening of the Ramsey property for relational structures, where the only colourings considered are those that are externally definable: that is, definable with parameters in an elementary extension. We show a number of basic results analogous to the classical Ramsey theory, and show that, for an ultrahomogeneous structure <i>M</i> with countable age, the externally definable Ramsey property is equivalent to the dynamical statement that, for all <span>(n in mathbb {N} )</span>, every subflow of the space <span>(S_n(M))</span> of <i>n</i>-types has a fixed point. We discuss a range of examples, including results regarding the lexicographic product of structures.\u0000</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"64 3-4","pages":"605 - 635"},"PeriodicalIF":0.3,"publicationDate":"2024-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143871444","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Bounded symbiosis and upwards reflection","authors":"Lorenzo Galeotti,&nbsp;Yurii Khomskii,&nbsp;Jouko Väänänen","doi":"10.1007/s00153-024-00955-0","DOIUrl":"10.1007/s00153-024-00955-0","url":null,"abstract":"<div><p>In Bagaria (J Symb Log 81(2), 584–604, 2016), Bagaria and Väänänen developed a framework for studying the large cardinal strength of <i>downwards</i> Löwenheim-Skolem theorems and related set theoretic reflection properties. The main tool was the notion of <i>symbiosis</i>, originally introduced by the third author in Väänänen (Applications of set theory to generalized quantifiers. PhD thesis, University of Manchester, 1967); Väänänen (in Logic Colloquium ’78 (Mons, 1978), volume 97 of Stud. Logic Foundations Math., pages 391–421. North-Holland, Amsterdam 1979) <i>Symbiosis</i> provides a way of relating model theoretic properties of strong logics to definability in set theory. In this paper we continue the systematic investigation of <i>symbiosis</i> and apply it to <i>upwards</i> Löwenheim-Skolem theorems and reflection principles. To achieve this, we need to adapt the notion of <i>symbiosis</i> to a new form, called <i>bounded symbiosis</i>. As one easy application, we obtain upper and lower bounds for the large cardinal strength of upwards Löwenheim–Skolem-type principles for second order logic.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"64 3-4","pages":"579 - 603"},"PeriodicalIF":0.3,"publicationDate":"2024-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00153-024-00955-0.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143871154","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Approximate categoricity in continuous logic","authors":"James E. Hanson","doi":"10.1007/s00153-024-00952-3","DOIUrl":"10.1007/s00153-024-00952-3","url":null,"abstract":"<div><p>We explore approximate categoricity in the context of distortion systems, introduced in our previous paper (Hanson in Math Logic Q 69(4):482–507, 2023), which are a mild generalization of perturbation systems, introduced by Yaacov (J Math Logic 08(02):225–249, 2008). We extend Ben Yaacov’s Ryll-Nardzewski style characterization of separably approximately categorical theories from the context of perturbation systems to that of distortion systems. We also make progress towards an analog of Morley’s theorem for inseparable approximate categoricity, showing that if there is some uncountable cardinal <span>(kappa )</span> such that every model of size <span>(kappa )</span> is ‘approximately saturated,’ in the appropriate sense, then the same is true for all uncountable cardinalities. Finally we present some examples of these phenomena and highlight an apparent interaction between ordinary separable categoricity and inseparable approximate categoricity.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"64 3-4","pages":"547 - 577"},"PeriodicalIF":0.3,"publicationDate":"2024-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143871398","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Variations on the Feferman-Vaught theorem, with applications to (prod _p mathbb {F}_p)","authors":"Alice Medvedev,&nbsp;Alexander Van Abel","doi":"10.1007/s00153-024-00954-1","DOIUrl":"10.1007/s00153-024-00954-1","url":null,"abstract":"<div><p>Using the Feferman-Vaught Theorem, we prove that a definable subset of a product structure must be a Boolean combination of open sets, in the product topology induced by giving each factor structure the discrete topology. We prove that for families of structures with certain properties, including families of integral domains, the pure Boolean generalized product is definable in the direct product structure. We use these results to obtain characterizations of the definable subsets of <span>(prod _p mathbb {F}_p)</span>—in particular, every formula is equivalent to a Boolean combination of <span>(exists forall exists )</span> formulae.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"64 3-4","pages":"529 - 546"},"PeriodicalIF":0.3,"publicationDate":"2024-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143871199","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the consistency strength of critical leaps","authors":"Gunter Fuchs","doi":"10.1007/s00153-024-00951-4","DOIUrl":"10.1007/s00153-024-00951-4","url":null,"abstract":"<div><p>In the analysis of the blurry <span>(textsf{HOD})</span> hierarchy, one of the fundamental concepts is that of a leap, and it turned out that critical leaps are of particular interest. A critical leap is a leap which is the cardinal successor of a singular strong limit cardinal. Such a leap is sudden if its cardinal predecessor is not a leap, and otherwise, it is smooth. In prior work, I showed that the existence of a sudden critical leap is equiconsistent with the existence of a measurable cardinal. Here, I show that if the cofinality of the cardinal predecessor of a sudden critical leap is required to be uncountable, the consistency strength increases considerably. I also show that when focusing on critical leaps whose cardinal predecessors have uncountable cofinality, the consistency strength of a smooth critical leap is much lower than that of a sudden critical leap. Finally, I observe that in contrast to the countable cofinality setting, <span>(aleph _{omega _1+1})</span>, e.g., cannot be a sudden critical leap.\u0000</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"64 3-4","pages":"515 - 528"},"PeriodicalIF":0.3,"publicationDate":"2024-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143871443","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Relativized Galois groups of first order theories over a hyperimaginary","authors":"Hyoyoon Lee,&nbsp;Junguk Lee","doi":"10.1007/s00153-024-00953-2","DOIUrl":"10.1007/s00153-024-00953-2","url":null,"abstract":"<div><p>We study relativized Lascar groups, which are formed by relativizing Lascar groups to the solution set of a partial type <span>(Sigma )</span>. We introduce the notion of a Lascar tuple for <span>(Sigma )</span> and by considering the space of types over a Lascar tuple for <span>(Sigma )</span>, the topology for a relativized Lascar group is (re-)defined and some fundamental facts about the Galois groups of first-order theories are generalized to the relativized context. In particular, we prove that any closed subgroup of a relativized Lascar group corresponds to a stabilizer of a bounded hyperimaginary having at least one representative in the solution set of the given partial type <span>(Sigma )</span>. Using this, we find the correspondence between subgroups of the relativized Lascar group and the relativized strong types.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"64 3-4","pages":"493 - 514"},"PeriodicalIF":0.3,"publicationDate":"2024-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143871305","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Infinite combinatorics revisited in the absence of Axiom of choice","authors":"Tamás Csernák,&nbsp;Lajos Soukup","doi":"10.1007/s00153-024-00946-1","DOIUrl":"10.1007/s00153-024-00946-1","url":null,"abstract":"&lt;div&gt;&lt;p&gt;We investigate whether classical combinatorial theorems are provable in ZF. Some statements are not provable in ZF, but they are equivalent within ZF. For example, the following statements (i)–(iii) are equivalent: &lt;/p&gt;&lt;ol&gt;\u0000 &lt;li&gt;\u0000 &lt;span&gt;(i)&lt;/span&gt;\u0000 \u0000 &lt;p&gt;&lt;span&gt;(cf({omega }_1)={omega }_1)&lt;/span&gt;,&lt;/p&gt;\u0000 \u0000 &lt;/li&gt;\u0000 &lt;li&gt;\u0000 &lt;span&gt;(ii)&lt;/span&gt;\u0000 \u0000 &lt;p&gt;&lt;span&gt;({omega }_1rightarrow ({omega }_1,{omega }+1)^2)&lt;/span&gt;,&lt;/p&gt;\u0000 \u0000 &lt;/li&gt;\u0000 &lt;li&gt;\u0000 &lt;span&gt;(iii)&lt;/span&gt;\u0000 \u0000 &lt;p&gt;any family &lt;span&gt;(mathcal {A}subset [{On}]^{&lt;{omega }})&lt;/span&gt; of size &lt;span&gt;({omega }_1)&lt;/span&gt; contains a &lt;span&gt;(Delta )&lt;/span&gt;-system of size &lt;span&gt;({omega }_1)&lt;/span&gt;.&lt;/p&gt;\u0000 \u0000 &lt;/li&gt;\u0000 &lt;/ol&gt;&lt;p&gt; Some classical results cannot be proven in ZF alone; however, we can establish weaker versions of these statements within the framework of ZF, such as &lt;/p&gt;&lt;ol&gt;\u0000 &lt;li&gt;\u0000 &lt;span&gt;(1)&lt;/span&gt;\u0000 \u0000 &lt;p&gt;&lt;span&gt;({{omega }_2}rightarrow ({omega }_1,{omega }+1))&lt;/span&gt;,&lt;/p&gt;\u0000 \u0000 &lt;/li&gt;\u0000 &lt;li&gt;\u0000 &lt;span&gt;(2)&lt;/span&gt;\u0000 \u0000 &lt;p&gt;any family &lt;span&gt;(mathcal {A}subset [{On}]^{&lt;{omega }})&lt;/span&gt; of size &lt;span&gt;({omega }_2)&lt;/span&gt; contains a &lt;span&gt;(Delta )&lt;/span&gt;-system of size &lt;span&gt;({omega }_1)&lt;/span&gt;.&lt;/p&gt;\u0000 \u0000 &lt;/li&gt;\u0000 &lt;/ol&gt;&lt;p&gt; Some statements can be proven in ZF using purely combinatorial arguments, such as: &lt;/p&gt;&lt;ol&gt;\u0000 &lt;li&gt;\u0000 &lt;span&gt;(3)&lt;/span&gt;\u0000 \u0000 &lt;p&gt;given a set mapping &lt;span&gt;(F:{omega }_1rightarrow {[{omega }_1]}^{&lt;{omega }})&lt;/span&gt;, the set &lt;span&gt;({omega }_1)&lt;/span&gt; has a partition into &lt;span&gt;({omega })&lt;/span&gt;-many &lt;i&gt;F&lt;/i&gt;-free sets.&lt;/p&gt;\u0000 \u0000 &lt;/li&gt;\u0000 &lt;/ol&gt;&lt;p&gt; Other statements can be proven in ZF by employing certain methods of absoluteness, for example: &lt;/p&gt;&lt;ol&gt;\u0000 &lt;li&gt;\u0000 &lt;span&gt;(4)&lt;/span&gt;\u0000 \u0000 &lt;p&gt;given a set mapping &lt;span&gt;(F:{omega }_1rightarrow {[{omega }_1]}^{&lt;{omega }})&lt;/span&gt;, there is an &lt;i&gt;F&lt;/i&gt;-free set of size &lt;span&gt;({omega }_1)&lt;/span&gt;,&lt;/p&gt;\u0000 \u0000 &lt;/li&gt;\u0000 &lt;li&gt;\u0000 &lt;span&gt;(5)&lt;/span&gt;\u0000 \u0000 &lt;p&gt;for each &lt;span&gt;(nin {omega })&lt;/span&gt;, every family &lt;span&gt;(mathcal {A}subset {[{omega }_1]}^{{omega }})&lt;/span&gt; with &lt;span&gt;(|Acap B|le n)&lt;/span&gt; for &lt;span&gt;({A,B}in {[mathcal {A}]}^{2})&lt;/span&gt; has property &lt;i&gt;B&lt;/i&gt;.&lt;/p&gt;\u0000 \u0000 &lt;/li&gt;\u0000 &lt;/ol&gt;&lt;p&gt; In contrast to statement (5), we show that the following ZFC theorem of Komjáth is not provabl","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"64 3-4","pages":"473 - 491"},"PeriodicalIF":0.3,"publicationDate":"2024-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00153-024-00946-1.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143871303","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"What would the rational Urysohn space and the random graph look like if they were uncountable?","authors":"Ziemowit Kostana","doi":"10.1007/s00153-024-00948-z","DOIUrl":"10.1007/s00153-024-00948-z","url":null,"abstract":"<div><p>Building on the work of Avraham, Rubin, and Shelah, we aim to build a variant of the Fraïssé theory for uncountable models built from finite submodels. With this aim, we generalize the notion of an increasing set of reals to other structures. As an application, we prove that the following is consistent: there exists an uncountable, separable metric space <i>X</i> with rational distances, such that every uncountable partial 1-1 function from <i>X</i> to <i>X</i> is an isometry on an uncountable subset. We aim for a general theory of structures with this kind of properties. This includes results about the automorphism groups, and partial classification results.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"64 3-4","pages":"445 - 472"},"PeriodicalIF":0.3,"publicationDate":"2024-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143871393","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Degrees of relations on canonically ordered natural numbers and integers","authors":"Nikolay Bazhenov,&nbsp;Dariusz Kalociński,&nbsp;Michał Wrocławski","doi":"10.1007/s00153-024-00942-5","DOIUrl":"10.1007/s00153-024-00942-5","url":null,"abstract":"<div><p>We investigate the degree spectra of computable relations on canonically ordered natural numbers <span>((omega ,&lt;))</span> and integers <span>((zeta ,&lt;))</span>. As for <span>((omega ,&lt;))</span>, we provide several criteria that fix the degree spectrum of a computable relation to all c.e. or to all <span>(Delta _2)</span> degrees; this includes the complete characterization of the degree spectra of so-called computable block functions that have only finitely many types of blocks. Compared to Bazhenov et al. (in: LIPIcs, vol 219, pp 8:1–8:20, 2022), we obtain a more general solution to the problem regarding possible degree spectra on <span>((omega ,&lt;))</span>, answering the question whether there are infinitely many such spectra. As for <span>((zeta ,&lt;))</span>, we prove the following dichotomy result: given an arbitrary computable relation <i>R</i> on <span>((zeta ,&lt;))</span>, its degree spectrum is either trivial or it contains all c.e. degrees. This result, and the proof techniques required to solve it, extend the analogous theorem for <span>((omega ,&lt;))</span> obtained by Wright (Computability 7:349–365, 2018), and provide initial insight to Wright’s question whether such a dichotomy holds on computable ill-founded linear orders. This article is an extended version of Bazhenov et al. (in: LIPIcs, vol 219, pp 8:1–8:20, 2022).</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"64 1-2","pages":"299 - 331"},"PeriodicalIF":0.3,"publicationDate":"2024-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00153-024-00942-5.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143388706","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Free subsets in internally approachable models","authors":"P. D. Welch","doi":"10.1007/s00153-024-00947-0","DOIUrl":"10.1007/s00153-024-00947-0","url":null,"abstract":"<div><p>We consider a question of Pereira as to whether the characteristic function of an internally approachable model can lead to free subsets for functions of the model. Pereira isolated the pertinent <i>Approachable Free Subsets Property</i> (AFSP) in his work on the <span>({text {pcf}})</span>-conjecture. A recent related property is the <i>Approachable Bounded Subset Property</i> (ABSP) of Ben-Neria and Adolf, and we here directly show it requires modest large cardinals to establish:</p><p><b>Theorem</b> <i>If ABSP</i> holds for an ascending sequence <span>( langle aleph _{n_{m}} rangle _{m})</span> <span>(( n_{m} in omega ))</span> then there is an inner model with measurables <span>(kappa &lt; aleph _{omega })</span> of arbitrarily large Mitchell order below <span>(aleph _{omega })</span>, that is: <span>(sup left{ alpha mid {exists }kappa &lt; aleph _{omega } o ( kappa ) ge alpha right} = aleph _{omega })</span>. A result of Adolf and Ben Neria then shows that this conclusion is in fact the exact consistency strength of ABSP for such an ascending sequence. Their result went via the consistency of the non-existence of continuous tree-like scales; the result of this paper is direct and avoids the use of PCF scales.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"64 3-4","pages":"435 - 443"},"PeriodicalIF":0.3,"publicationDate":"2024-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00153-024-00947-0.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143871401","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}