Jörg Brendle , Lorenz J. Halbeisen , Lukas Daniel Klausner , Marc Lischka , Saharon Shelah
{"title":"Halfway new cardinal characteristics","authors":"Jörg Brendle , Lorenz J. Halbeisen , Lukas Daniel Klausner , Marc Lischka , Saharon Shelah","doi":"10.1016/j.apal.2023.103303","DOIUrl":"https://doi.org/10.1016/j.apal.2023.103303","url":null,"abstract":"<div><p>Based on the well-known cardinal characteristics <span><math><mi>s</mi></math></span>, <span><math><mi>r</mi></math></span> and <span><math><mi>i</mi></math></span>, we introduce nine related cardinal characteristics by using the notion of asymptotic density to characterise different intersection properties of infinite sets. We prove several bounds and consistency results, e.g. the consistency of <span><math><mi>s</mi><mo><</mo><msub><mrow><mi>s</mi></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msub></math></span>, as well as several results about possible values of <span><math><msub><mrow><mi>i</mi></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msub></math></span>.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"174 9","pages":"Article 103303"},"PeriodicalIF":0.8,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49754699","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}
Nikolay Bazhenov , Matthew Harrison-Trainor , Alexander Melnikov
{"title":"Computable Stone spaces","authors":"Nikolay Bazhenov , Matthew Harrison-Trainor , Alexander Melnikov","doi":"10.1016/j.apal.2023.103304","DOIUrl":"https://doi.org/10.1016/j.apal.2023.103304","url":null,"abstract":"<div><p>We investigate computable metrizability of Polish spaces up to homeomorphism. In this paper we focus on Stone spaces. We use Stone duality to construct the first known example of a computable topological Polish space not homeomorphic to any computably metrized space. In fact, in our proof we construct a right-c.e. metrized Stone space which is not homeomorphic to any computably metrized space. Then we introduce a new notion of effective categoricity for effectively compact spaces and prove that effectively categorical Stone spaces are exactly the duals of computably categorical Boolean algebras. Finally, we prove that, for a Stone space <em>X</em>, the Banach space <span><math><mi>C</mi><mo>(</mo><mi>X</mi><mo>;</mo><mi>R</mi><mo>)</mo></math></span> has a computable presentation if, and only if, <em>X</em> is <em>homeomorphic</em> to a computably metrized space. This gives an unexpected positive partial answer to a question recently posed by McNicholl.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"174 9","pages":"Article 103304"},"PeriodicalIF":0.8,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49754700","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":"Fresh function spectra","authors":"Vera Fischer, Marlene Koelbing, Wolfgang Wohofsky","doi":"10.1016/j.apal.2023.103300","DOIUrl":"https://doi.org/10.1016/j.apal.2023.103300","url":null,"abstract":"<div><p>In this paper, we investigate the fresh function spectrum of forcing notions, where a new function on an ordinal is called fresh if all its initial segments are in the ground model. We determine the fresh function spectrum of several forcing notions and discuss the difference between fresh functions and fresh subsets. Furthermore, we consider the question which sets are realizable as the fresh function spectrum of a homogeneous forcing. We show that under GCH all sets with a certain closure property are realizable, while consistently there are sets which are not realizable.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"174 9","pages":"Article 103300"},"PeriodicalIF":0.8,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49738585","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 a finer classification of strongly minimal sets","authors":"John T. Baldwin , Viktor V. Verbovskiy","doi":"10.1016/j.apal.2023.103376","DOIUrl":"https://doi.org/10.1016/j.apal.2023.103376","url":null,"abstract":"<div><p>Let <em>M</em> be strongly minimal and constructed by a ‘Hrushovski construction’ with a single ternary relation. If the Hrushovski algebraization function <em>μ</em> is in a certain class <span><math><mi>T</mi></math></span> (<em>μ</em> triples) we show that for independent <em>I</em> with <span><math><mo>|</mo><mi>I</mi><mo>|</mo><mo>></mo><mn>1</mn></math></span>, <span><math><msup><mrow><mi>dcl</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>(</mo><mi>I</mi><mo>)</mo><mo>=</mo><mo>∅</mo></math></span> (* means not in dcl of a proper subset). This implies the only definable truly <em>n</em>-ary functions <em>f</em> (<em>f</em> ‘depends’ on each argument), occur when <span><math><mi>n</mi><mo>=</mo><mn>1</mn></math></span>. We prove for Hrushovski's original construction and for the strongly minimal <em>k</em>-Steiner systems of Baldwin and Paolini that the symmetric definable closure, <span><math><msup><mrow><mi>sdcl</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>(</mo><mi>I</mi><mo>)</mo><mo>=</mo><mo>∅</mo></math></span> (<span>Definition 2.7</span>). Thus, no such theory admits elimination of imaginaries. As, we show that in an arbitrary strongly minimal theory, elimination of imaginaries implies <span><math><msup><mrow><mi>sdcl</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>(</mo><mi>I</mi><mo>)</mo><mo>≠</mo><mo>∅</mo></math></span>. In particular, such strongly minimal Steiner systems with line-length at least 4 do not interpret a quasigroup, even though they admit a coordinatization if <span><math><mi>k</mi><mo>=</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>. The case structure depends on properties of the Hrushovski <em>μ</em>-function. The proofs depend on our introduction, for appropriate <span><math><mi>G</mi><mo>⊆</mo><mrow><mi>aut</mi></mrow><mo>(</mo><mi>M</mi><mo>)</mo></math></span> (setwise or pointwise stabilizers of finite independent sets), the notion of a <em>G</em>-normal substructure <span><math><mi>A</mi></math></span> of <em>M</em> and of a <em>G</em>-decomposition of any finite such <span><math><mi>A</mi></math></span>. These results lead to a finer classification of strongly minimal structures with flat geometry, according to what sorts of definable functions they admit.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"175 2","pages":"Article 103376"},"PeriodicalIF":0.8,"publicationDate":"2023-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49737987","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":"Classification of ℵ0-categorical C-minimal pure C-sets","authors":"Françoise Delon , Marie-Hélène Mourgues","doi":"10.1016/j.apal.2023.103375","DOIUrl":"https://doi.org/10.1016/j.apal.2023.103375","url":null,"abstract":"<div><p>We classify all <span><math><msub><mrow><mi>ℵ</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>-categorical and <em>C</em>-minimal <em>C</em>-sets up to elementary equivalence. As usual the Ryll-Nardzewski Theorem makes the classification of indiscernible <span><math><msub><mrow><mi>ℵ</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>-categorical <em>C</em>-minimal sets as a first step. We first define <em>solvable</em> good trees, via a finite induction. The trees involved in initial and induction steps have a set of nodes, either consisting of a singleton, or having dense branches without endpoints and the same number of branches at each node. The class of <em>colored</em> good trees is the elementary class of solvable good trees. We show that a pure <em>C</em>-set <em>M</em> is indiscernible, finite or <span><math><msub><mrow><mi>ℵ</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>-categorical and <em>C</em>-minimal iff its canonical tree <span><math><mi>T</mi><mo>(</mo><mi>M</mi><mo>)</mo></math></span> is a colored good tree. The classification of general <span><math><msub><mrow><mi>ℵ</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>-categorical and <em>C</em>-minimal <em>C</em>-sets is done via finite trees with labeled vertices and edges, where labels are natural numbers, or infinity and complete theories of indiscernible, <span><math><msub><mrow><mi>ℵ</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>-categorical or finite, and <em>C</em>-minimal <em>C</em>-sets.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"175 2","pages":"Article 103375"},"PeriodicalIF":0.8,"publicationDate":"2023-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49725104","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}
Nick Bezhanishvili , Anna Dmitrieva , Jim de Groot , Tommaso Moraschini
{"title":"Positive modal logic beyond distributivity","authors":"Nick Bezhanishvili , Anna Dmitrieva , Jim de Groot , Tommaso Moraschini","doi":"10.1016/j.apal.2023.103374","DOIUrl":"https://doi.org/10.1016/j.apal.2023.103374","url":null,"abstract":"<div><p>We develop a duality for (modal) lattices that need not be distributive, and use it to study positive (modal) logic beyond distributivity, which we call weak positive (modal) logic. This duality builds on the Hofmann, Mislove and Stralka duality for meet-semilattices. We introduce the notion of <span><math><msub><mrow><mi>Π</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>-persistence and show that every weak positive modal logic is <span><math><msub><mrow><mi>Π</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>-persistent. This approach leads to a new relational semantics for weak positive modal logic, for which we prove an analogue of Sahlqvist's correspondence result.<span><sup>1</sup></span></p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"175 2","pages":"Article 103374"},"PeriodicalIF":0.8,"publicationDate":"2023-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49737990","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":"Weak saturation properties and side conditions","authors":"Monroe Eskew","doi":"10.1016/j.apal.2023.103356","DOIUrl":"10.1016/j.apal.2023.103356","url":null,"abstract":"<div><p>Towards combining “compactness” and “hugeness” properties at <span><math><msub><mrow><mi>ω</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>, we investigate the relevance of side-conditions forcing. We reduce the upper bound on the consistency strength of the weak Chang's Conjecture at <span><math><msub><mrow><mi>ω</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> using Neeman's forcing. On the other hand, we find a barrier to the applicability of these methods to our problem and give a counterexample to a claim of Neeman about the effects of iterating such forcing.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"175 1","pages":"Article 103356"},"PeriodicalIF":0.8,"publicationDate":"2023-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42596957","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":"Absoluteness for the theory of the inner model constructed from finitely many cofinality quantifiers","authors":"Ur Ya'ar","doi":"10.1016/j.apal.2023.103358","DOIUrl":"10.1016/j.apal.2023.103358","url":null,"abstract":"<div><p>We prove that the theory of the models constructible using finitely many cofinality quantifiers – <span><math><msubsup><mrow><mi>C</mi></mrow><mrow><msub><mrow><mi>λ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow><mrow><mo>⁎</mo></mrow></msubsup></math></span> and <span><math><msubsup><mrow><mi>C</mi></mrow><mrow><mo><</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><mo><</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow><mrow><mo>⁎</mo></mrow></msubsup></math></span> for <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> regular cardinals – is set-forcing absolute under the assumption of class many Woodin cardinals, and is independent of the regular cardinals used. Towards this goal we prove some properties of the generic embedding induced from the stationary tower restricted to <<em>μ</em>-closed sets.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"175 1","pages":"Article 103358"},"PeriodicalIF":0.8,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45405170","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}
Alexander I. Bufetov , Gabriel Nivasch , Fedor Pakhomov
{"title":"Generalized fusible numbers and their ordinals","authors":"Alexander I. Bufetov , Gabriel Nivasch , Fedor Pakhomov","doi":"10.1016/j.apal.2023.103355","DOIUrl":"https://doi.org/10.1016/j.apal.2023.103355","url":null,"abstract":"<div><p>Erickson defined the <em>fusible numbers</em> as a set <span><math><mi>F</mi></math></span> of reals generated by repeated application of the function <span><math><mfrac><mrow><mi>x</mi><mo>+</mo><mi>y</mi><mo>+</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></math></span>. Erickson, Nivasch, and Xu showed that <span><math><mi>F</mi></math></span> is well ordered, with order type <span><math><msub><mrow><mi>ε</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>. They also investigated a recursively defined function <span><math><mi>M</mi><mo>:</mo><mi>R</mi><mo>→</mo><mi>R</mi></math></span>. They showed that the set of points of discontinuity of <em>M</em> is a subset of <span><math><mi>F</mi></math></span> of order type <span><math><msub><mrow><mi>ε</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>. They also showed that, although <em>M</em> is a total function on <span><math><mi>R</mi></math></span>, the fact that the restriction of <em>M</em> to <span><math><mi>Q</mi></math></span> is total is not provable in first-order Peano arithmetic <span><math><mi>PA</mi></math></span>.</p><p>In this paper we explore the problem (raised by Friedman) of whether similar approaches can yield well-ordered sets <span><math><mi>F</mi></math></span> of larger order types. As Friedman pointed out, Kruskal's tree theorem yields an upper bound of the small Veblen ordinal for the order type of any set generated in a similar way by repeated application of a monotone function <span><math><mi>g</mi><mo>:</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>→</mo><mi>R</mi></math></span>.</p><p>The most straightforward generalization of <span><math><mfrac><mrow><mi>x</mi><mo>+</mo><mi>y</mi><mo>+</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></math></span> to an <em>n</em>-ary function is the function <span><math><mfrac><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>+</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></mfrac></math></span>. We show that this function generates a set <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> whose order type is just <span><math><msub><mrow><mi>φ</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>(</mo><mn>0</mn><mo>)</mo></math></span>. For this, we develop recursively defined functions <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>:</mo><mi>R</mi><mo>→</mo><mi>R</mi></math></span> naturally generalizing the function <em>M</em>.</p><p>Furthermore, we prove that for any <em>linear</em> function <span><math><mi>g</mi><mo>:</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>→</mo><mi>R</mi></math></span>, the order type of the resulting <span><math><mi>F</mi></math></span> is at most <span><math><msub><mrow><mi>φ</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>(</mo><mn>0</mn><","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"175 1","pages":"Article 103355"},"PeriodicalIF":0.8,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49724785","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":"Extensions of Solovay's system S without independent sets of axioms","authors":"Igor Gorbunov , Dmitry Shkatov","doi":"10.1016/j.apal.2023.103360","DOIUrl":"10.1016/j.apal.2023.103360","url":null,"abstract":"<div><p>Chagrov and Zakharyaschev posed the problem of existence of extensions of Solovay's system <strong>S</strong>, which is a non-normalizable quasi-normal modal logic, that do not admit deductively independent sets of axioms. This paper gives a solution by exhibiting countably many extensions of <strong>S</strong> without deductively independent sets of axioms.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"175 1","pages":"Article 103360"},"PeriodicalIF":0.8,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45257728","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}
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