Annals of Pure and Applied Logic最新文献

{"title":"Unreachability of Γ2n+1,m","authors":"Derek Levinson","doi":"10.1016/j.apal.2025.103604","DOIUrl":"10.1016/j.apal.2025.103604","url":null,"abstract":"<div><div>We find bounds for the maximal length of a sequence of distinct <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mn>2n</mn><mo>+</mo><mn>1</mn><mo>,</mo><mi>m</mi></mrow></msub></math></span>-sets under <em>AD</em> and show there is no sequence of distinct <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mn>2n</mn><mo>+</mo><mn>1</mn></mrow></msub></math></span>-sets of length <span><math><msubsup><mrow><mi>δ</mi></mrow><mrow><mn>2n</mn><mo>+</mo><mn>3</mn></mrow><mrow><mn>1</mn></mrow></msubsup></math></span>. As a special case, there is no sequence of distinct <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>m</mi></mrow></msub></math></span>-sets of length <span><math><msub><mrow><mi>ℵ</mi></mrow><mrow><mi>m</mi><mo>+</mo><mn>2</mn></mrow></msub></math></span>. These are the optimal results for the pointclasses <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mn>2n</mn><mo>+</mo><mn>1</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>m</mi></mrow></msub></math></span>.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 9","pages":"Article 103604"},"PeriodicalIF":0.6,"publicationDate":"2025-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143895035","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":"Metric spaces in choiceless set theory","authors":"Eleftherios Tachtsis","doi":"10.1016/j.apal.2025.103603","DOIUrl":"10.1016/j.apal.2025.103603","url":null,"abstract":"&lt;div&gt;&lt;div&gt;We &lt;em&gt;answer open questions&lt;/em&gt; from Keremedis (2016) &lt;span&gt;&lt;span&gt;[12]&lt;/span&gt;&lt;/span&gt; and Keremedis and Tachtsis (2022) &lt;span&gt;&lt;span&gt;[16]&lt;/span&gt;&lt;/span&gt;, and &lt;em&gt;properly strengthen some results&lt;/em&gt; from the above papers as well as from Keremedis et al. (2023) &lt;span&gt;&lt;span&gt;[19]&lt;/span&gt;&lt;/span&gt;. In particular, and among other results, we establish the following:&lt;ul&gt;&lt;li&gt;&lt;span&gt;1.&lt;/span&gt;&lt;span&gt;&lt;div&gt;The Boolean Prime Ideal Theorem does not imply “For every sequentially compact metric space &lt;span&gt;&lt;math&gt;&lt;mo&gt;〈&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mo&gt;〉&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;ℵ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;” in &lt;strong&gt;ZF&lt;/strong&gt; (Zermelo–Fraenkel set theory without the Axiom of Choice (&lt;strong&gt;AC&lt;/strong&gt;)).&lt;/div&gt;&lt;/span&gt;&lt;/li&gt;&lt;li&gt;&lt;span&gt;2.&lt;/span&gt;&lt;span&gt;&lt;div&gt;“Every linearly ordered set can be well ordered” ∧ “The union of a well-orderable family of well-orderable sets is well orderable” ∧ “For every uncountable sequentially compact metric space &lt;span&gt;&lt;math&gt;&lt;mo&gt;〈&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mo&gt;〉&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;ℵ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;” does not imply the Axiom of Countable Choice in &lt;strong&gt;ZFA&lt;/strong&gt; (&lt;strong&gt;ZF&lt;/strong&gt; with atoms).&lt;/div&gt;&lt;/span&gt;&lt;/li&gt;&lt;li&gt;&lt;span&gt;3.&lt;/span&gt;&lt;span&gt;&lt;div&gt;“For every uncountable compact metric space &lt;span&gt;&lt;math&gt;&lt;mo&gt;〈&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mo&gt;〉&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;ℵ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;” does not imply “For every uncountable sequentially compact metric space &lt;span&gt;&lt;math&gt;&lt;mo&gt;〈&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mo&gt;〉&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;ℵ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;” in &lt;strong&gt;ZFA&lt;/strong&gt;”.&lt;/div&gt;&lt;/span&gt;&lt;/li&gt;&lt;li&gt;&lt;span&gt;4.&lt;/span&gt;&lt;span&gt;&lt;div&gt;“For every uncountable sequentially compact metric space &lt;span&gt;&lt;math&gt;&lt;mo&gt;〈&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mo&gt;〉&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; &lt;span&gt;&lt;math&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;ℵ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;” does not imply “For every uncountable compact metric space &lt;span&gt;&lt;math&gt;&lt;mo&gt;〈&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mo&gt;〉&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;ℵ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;” in &lt;strong&gt;ZFA&lt;/strong&gt;.&lt;/div&gt;&lt;/span&gt;&lt;/li&gt;&lt;li&gt;&lt;span&gt;5.&lt;/span&gt;&lt;span&gt;&lt;div&gt;“For every uncountable sequentially compact metric space &lt;span&gt;&lt;math&gt;&lt;mo&gt;〈&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mo&gt;","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 9","pages":"Article 103603"},"PeriodicalIF":0.6,"publicationDate":"2025-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143886674","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":"Completeness in local positive logic","authors":"Arturo Rodríguez Fanlo ,&nbsp;Ori Segel","doi":"10.1016/j.apal.2025.103601","DOIUrl":"10.1016/j.apal.2025.103601","url":null,"abstract":"<div><div>We develop the basic model theory of local positive logic, a new logic that mixes positive logic (where negation is not allowed) and local logic (where models omit types of infinite distant pairs). We study several basic model theoretic notions such as compactness, positive closedness (existential closedness) and completeness (irreducibility).</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 7","pages":"Article 103601"},"PeriodicalIF":0.6,"publicationDate":"2025-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143820322","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":"The Set-Cover game and non-measurable unions","authors":"Taras Banakh ,&nbsp;Robert Rałowski ,&nbsp;Szymon Żeberski","doi":"10.1016/j.apal.2025.103602","DOIUrl":"10.1016/j.apal.2025.103602","url":null,"abstract":"<div><div>Using a game-theoretic approach we present a generalization of the classical result of Brzuchowski, Cichoń, Grzegorek and Ryll-Nardzewski on non-measurable unions. We also present applications of obtained results to Marczewski–Burstin representable ideals.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 8","pages":"Article 103602"},"PeriodicalIF":0.6,"publicationDate":"2025-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143808427","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":"Some more results on relativized Chaitin's Ω","authors":"Liang Yu","doi":"10.1016/j.apal.2025.103586","DOIUrl":"10.1016/j.apal.2025.103586","url":null,"abstract":"<div><div>We prove that, assuming ZF, and restricted to any <span><math><msub><mrow><mo>≤</mo></mrow><mrow><mi>T</mi></mrow></msub></math></span>-pointed set, Chaitin's <span><math><msub><mrow><mi>Ω</mi></mrow><mrow><mi>U</mi></mrow></msub><mo>:</mo><mi>x</mi><mo>↦</mo><msubsup><mrow><mi>Ω</mi></mrow><mrow><mi>U</mi></mrow><mrow><mi>x</mi></mrow></msubsup><mo>=</mo><msub><mrow><mo>∑</mo></mrow><mrow><msup><mrow><mi>U</mi></mrow><mrow><mi>x</mi></mrow></msup><mo>(</mo><mi>σ</mi><mo>)</mo><mo>↓</mo></mrow></msub><msup><mrow><mn>2</mn></mrow><mrow><mo>−</mo><mo>|</mo><mi>σ</mi><mo>|</mo></mrow></msup></math></span> is not injective for any universal prefix-free Turing machine <em>U</em>, and that <span><math><msubsup><mrow><mi>Ω</mi></mrow><mrow><mi>U</mi></mrow><mrow><mi>x</mi></mrow></msubsup></math></span> fails to be degree invariant in a very strong sense, answering several recent questions in descriptive set theory. Moreover, we show that under <span><math><mrow><mi>ZF</mi></mrow><mo>+</mo><mrow><mi>AD</mi></mrow></math></span>, every function <em>f</em> mapping <em>x</em> to <em>x</em>-random must be uncountable-to-one over an upper cone of Turing degrees.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 8","pages":"Article 103586"},"PeriodicalIF":0.6,"publicationDate":"2025-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143816774","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":"Generics in invariant subsets of the group of order preserving permutations of Q","authors":"M. Drzewiecka ,&nbsp;A. Ivanov ,&nbsp;B. Mokry","doi":"10.1016/j.apal.2025.103585","DOIUrl":"10.1016/j.apal.2025.103585","url":null,"abstract":"<div><div>Let <span><math><mi>ρ</mi><mo>∈</mo><mrow><mi>Aut</mi></mrow><mo>(</mo><mi>Q</mi><mo>,</mo><mo>&lt;</mo><mo>)</mo></math></span> and <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>ρ</mi></mrow></msub></math></span> be the closure of the conjugacy class of <em>ρ</em> in <span><math><mrow><mi>Aut</mi></mrow><mo>(</mo><mi>Q</mi><mo>,</mo><mo>&lt;</mo><mo>)</mo></math></span>. We show that <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>ρ</mi></mrow></msub></math></span> contains a conjugacy class, say <em>C</em>, which is comeagre in <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>ρ</mi></mrow></msub></math></span>. We describe representatives of <em>C</em>. Furthermore, we show that the family of finite partial maps extendable to elements of <em>C</em> has the cofinal amalgamation property.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 8","pages":"Article 103585"},"PeriodicalIF":0.6,"publicationDate":"2025-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143799916","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":"Extender-based Magidor-Radin forcings without top extenders","authors":"Moti Gitik ,&nbsp;Sittinon Jirattikansakul","doi":"10.1016/j.apal.2025.103584","DOIUrl":"10.1016/j.apal.2025.103584","url":null,"abstract":"<div><div>Continuing <span><span>[1]</span></span>, we develop a version of Extender-based Magidor-Radin forcing where there are no extenders on the top ordinal. As an application, we provide another approach to obtain a failure of SCH on a club subset of an inaccessible cardinal, and a model where the cardinal arithmetic behaviors are different on stationary classes, whose union is the club, is provided. The cardinals and the cofinalities outside the clubs are not affected by the forcings.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 8","pages":"Article 103584"},"PeriodicalIF":0.6,"publicationDate":"2025-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143760627","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":"Upward Löwenheim-Skolem-Tarski numbers for abstract logics","authors":"Victoria Gitman ,&nbsp;Jonathan Osinski","doi":"10.1016/j.apal.2025.103583","DOIUrl":"10.1016/j.apal.2025.103583","url":null,"abstract":"<div><div>Galeotti, Khomskii and Väänänen recently introduced the notion of the upward Löwenheim-Skolem-Tarski number for a logic, strengthening the classical notion of a Hanf number. A cardinal <em>κ</em> is the <em>upward Löwenheim-Skolem-Tarski number</em> (ULST <em>number</em>) of a logic <span><math><mi>L</mi></math></span> if it is the least cardinal with the property that whenever <em>M</em> is a model of size at least <em>κ</em> satisfying a sentence <em>φ</em> in <span><math><mi>L</mi></math></span>, then there are arbitrarily large models satisfying <em>φ</em> and having <em>M</em> as a substructure. The substructure requirement is what differentiates the ULST number from the Hanf number and gives the notion large cardinal strength. While it is a theorem of ZFC that every logic has a Hanf number, Galeotti, Khomskii and Väänänen showed that the existence of the ULST number for second-order logic implies the existence of a partially extendible cardinal. We answer positively their conjecture that the ULST number for second-order logic is the least extendible cardinal.</div><div>We define the <em>strong</em> ULST number by strengthening the substructure requirement to elementary substructure. We investigate the ULST and strong ULST numbers for several classical strong logics: infinitary logics, the equicardinality logic, logic with the well-foundedness quantifier, second-order logic, and sort logics. We show that the ULST and the strong ULST numbers are characterized in some cases by classical large cardinals and in some cases by natural new large cardinal notions that they give rise to. We show that for some logics the notions of the ULST number, strong ULST number and least strong compactness cardinal coincide, while for others, it is consistent that they can be separated. Finally, we introduce a natural large cardinal notion characterizing strong compactness cardinals for the equicardinality logic.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 8","pages":"Article 103583"},"PeriodicalIF":0.6,"publicationDate":"2025-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143748012","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":"Piecewise convex embeddability on linear orders","authors":"Martina Iannella ,&nbsp;Alberto Marcone ,&nbsp;Luca Motto Ros ,&nbsp;Vadim Weinstein","doi":"10.1016/j.apal.2025.103581","DOIUrl":"10.1016/j.apal.2025.103581","url":null,"abstract":"<div><div>Given a nonempty set <span><math><mi>L</mi></math></span> of linear orders, we say that the linear order <em>L</em> is <span><math><mi>L</mi></math></span>-convex embeddable into the linear order <span><math><msup><mrow><mi>L</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> if it is possible to partition <em>L</em> into convex sets indexed by some element of <span><math><mi>L</mi></math></span> which are isomorphic to convex subsets of <span><math><msup><mrow><mi>L</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> ordered in the same way. This notion generalizes convex embeddability and (finite) piecewise convex embeddability (both studied in <span><span>[13]</span></span>), which are the special cases <span><math><mi>L</mi><mo>=</mo><mo>{</mo><mn>1</mn><mo>}</mo></math></span> and <span><math><mi>L</mi><mo>=</mo><mrow><mi>Fin</mi></mrow></math></span>. We focus mainly on the behavior of these relations on the set of countable linear orders, first characterizing when they are transitive, and hence a quasi-order. We then study these quasi-orders from a combinatorial point of view, and analyze their complexity with respect to Borel reducibility. Finally, we extend our analysis to uncountable linear orders.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 8","pages":"Article 103581"},"PeriodicalIF":0.6,"publicationDate":"2025-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143769065","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":"Cardinal characteristics on bounded generalised Baire spaces","authors":"Tristan van der Vlugt","doi":"10.1016/j.apal.2025.103582","DOIUrl":"10.1016/j.apal.2025.103582","url":null,"abstract":"<div><div>We will give an overview of four families of cardinal characteristics defined on subspaces <span><math><msub><mrow><mo>∏</mo></mrow><mrow><mi>α</mi><mo>∈</mo><mi>κ</mi></mrow></msub><mi>b</mi><mo>(</mo><mi>α</mi><mo>)</mo></math></span> of the generalised Baire space <span><math><mmultiscripts><mrow><mi>κ</mi></mrow><mprescripts></mprescripts><none></none><mrow><mi>κ</mi></mrow></mmultiscripts></math></span>, where <em>κ</em> is strongly inaccessible and <span><math><mi>b</mi><mo>∈</mo><msup><mrow></mrow><mrow><mi>κ</mi></mrow></msup><mi>κ</mi></math></span>. The considered families are bounded versions of the dominating, eventual difference, localisation and antilocalisation numbers, and their dual cardinals. We investigate parameters for which these cardinals are nontrivial and how the cardinals relate to each other and to other cardinals of the generalised Cichoń diagram. Finally we prove that different choices of parameters may lead to consistently distinct cardinals.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 7","pages":"Article 103582"},"PeriodicalIF":0.6,"publicationDate":"2025-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143686230","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}