Pub Date : 2025-04-22DOI: 10.1016/j.apal.2025.103603
Eleftherios Tachtsis
<div><div>We <em>answer open questions</em> from Keremedis (2016) <span><span>[12]</span></span> and Keremedis and Tachtsis (2022) <span><span>[16]</span></span>, and <em>properly strengthen some results</em> from the above papers as well as from Keremedis et al. (2023) <span><span>[19]</span></span>. In particular, and among other results, we establish the following:<ul><li><span>1.</span><span><div>The Boolean Prime Ideal Theorem does not imply “For every sequentially compact metric space <span><math><mo>〈</mo><mi>X</mi><mo>,</mo><mi>d</mi><mo>〉</mo></math></span>, <span><math><mo>|</mo><mi>X</mi><mo>|</mo><mo>≤</mo><msup><mrow><mn>2</mn></mrow><mrow><msub><mrow><mi>ℵ</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></msup></math></span>” in <strong>ZF</strong> (Zermelo–Fraenkel set theory without the Axiom of Choice (<strong>AC</strong>)).</div></span></li><li><span>2.</span><span><div>“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 <span><math><mo>〈</mo><mi>X</mi><mo>,</mo><mi>d</mi><mo>〉</mo></math></span>, <span><math><mo>|</mo><mi>X</mi><mo>|</mo><mo>=</mo><msup><mrow><mn>2</mn></mrow><mrow><msub><mrow><mi>ℵ</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></msup></math></span>” does not imply the Axiom of Countable Choice in <strong>ZFA</strong> (<strong>ZF</strong> with atoms).</div></span></li><li><span>3.</span><span><div>“For every uncountable compact metric space <span><math><mo>〈</mo><mi>X</mi><mo>,</mo><mi>d</mi><mo>〉</mo></math></span>, <span><math><mo>|</mo><mi>X</mi><mo>|</mo><mo>≥</mo><msup><mrow><mn>2</mn></mrow><mrow><msub><mrow><mi>ℵ</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></msup></math></span>” does not imply “For every uncountable sequentially compact metric space <span><math><mo>〈</mo><mi>X</mi><mo>,</mo><mi>d</mi><mo>〉</mo></math></span>, <span><math><mo>|</mo><mi>X</mi><mo>|</mo><mo>≥</mo><msup><mrow><mn>2</mn></mrow><mrow><msub><mrow><mi>ℵ</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></msup></math></span>” in <strong>ZFA</strong>”.</div></span></li><li><span>4.</span><span><div>“For every uncountable sequentially compact metric space <span><math><mo>〈</mo><mi>X</mi><mo>,</mo><mi>d</mi><mo>〉</mo></math></span> <span><math><mo>|</mo><mi>X</mi><mo>|</mo><mo>≥</mo><msup><mrow><mn>2</mn></mrow><mrow><msub><mrow><mi>ℵ</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></msup></math></span>” does not imply “For every uncountable compact metric space <span><math><mo>〈</mo><mi>X</mi><mo>,</mo><mi>d</mi><mo>〉</mo></math></span>, <span><math><mo>|</mo><mi>X</mi><mo>|</mo><mo>=</mo><msup><mrow><mn>2</mn></mrow><mrow><msub><mrow><mi>ℵ</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></msup></math></span>” in <strong>ZFA</strong>.</div></span></li><li><span>5.</span><span><div>“For every uncountable sequentially compact metric space <span><math><mo>〈</mo><mi>X</mi><mo>,</mo><mi>d</mi><mo>
我们回答了Keremedis(2016)[12]和Keremedis and Tachtsis(2022)[19]的开放性问题，并适当加强了上述论文以及Keremedis等人（2023）[19]的一些结果。特别地，在其他结果中，我们确立了以下几点：在ZF （Zermelo-Fraenkel集合论，无选择公理（AC））.2中，布尔素数理想定理并不意味着“对于每一个序紧度量空间< X,d >, |X|≤2¹0”。“每个线性有序集都可以良序”∧良序集族的并是良序的“∧”对于每一个不可数序紧度量空间< X,d >, |X|=2¹0”并不意味着ZFA（含原子的ZF）中的可数选择公理。“对于每一个不可数紧度量空间< X,d >, |X|≥2¹0”并不意味着在ZFA中“对于每一个不可数连续紧度量空间< X,d >, |X|≥2¹0”。ZFA.5中的“对于每一个不可数连续紧度量空间< X,d > |X|≥2¹0”并不意味着“对于每一个不可数紧度量空间< X,d >, |X|=2¹0”。“对于每一个不可数序紧度量空间< X,d > |X|≥2 ^ 0”并不蕴涵ZFA.6中可数集的可数选择公理（CMCω）。“每一个线性有序的集合都是良序的”并不意味着ZFA中的“有限集合的可数选择公理”∨“每一个无限紧度量空间都有一个无限分散的子空间”。我们还讨论了斯通定理“每个度量空间都是准紧的”（ST）的演绎强度的开放问题，并提供了ST是否意味着CMCω的非平凡部分答案。特别地，我们证明了形式上较弱的命题“对于每一个度量空间< X,d >， |X|≥2 ^ 0或< X,d >是准紧的”并不意味着在ZFA中有CMCω。我们还证明，对于每一个不可数正则基数κ，在ZF中的“for all infinite良序基数λ<；κ， the Principle of Dependent Choices for λ成立”并不隐含上述ST的弱形式。
{"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":"<div><div>We <em>answer open questions</em> from Keremedis (2016) <span><span>[12]</span></span> and Keremedis and Tachtsis (2022) <span><span>[16]</span></span>, and <em>properly strengthen some results</em> from the above papers as well as from Keremedis et al. (2023) <span><span>[19]</span></span>. In particular, and among other results, we establish the following:<ul><li><span>1.</span><span><div>The Boolean Prime Ideal Theorem does not imply “For every sequentially compact metric space <span><math><mo>〈</mo><mi>X</mi><mo>,</mo><mi>d</mi><mo>〉</mo></math></span>, <span><math><mo>|</mo><mi>X</mi><mo>|</mo><mo>≤</mo><msup><mrow><mn>2</mn></mrow><mrow><msub><mrow><mi>ℵ</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></msup></math></span>” in <strong>ZF</strong> (Zermelo–Fraenkel set theory without the Axiom of Choice (<strong>AC</strong>)).</div></span></li><li><span>2.</span><span><div>“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 <span><math><mo>〈</mo><mi>X</mi><mo>,</mo><mi>d</mi><mo>〉</mo></math></span>, <span><math><mo>|</mo><mi>X</mi><mo>|</mo><mo>=</mo><msup><mrow><mn>2</mn></mrow><mrow><msub><mrow><mi>ℵ</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></msup></math></span>” does not imply the Axiom of Countable Choice in <strong>ZFA</strong> (<strong>ZF</strong> with atoms).</div></span></li><li><span>3.</span><span><div>“For every uncountable compact metric space <span><math><mo>〈</mo><mi>X</mi><mo>,</mo><mi>d</mi><mo>〉</mo></math></span>, <span><math><mo>|</mo><mi>X</mi><mo>|</mo><mo>≥</mo><msup><mrow><mn>2</mn></mrow><mrow><msub><mrow><mi>ℵ</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></msup></math></span>” does not imply “For every uncountable sequentially compact metric space <span><math><mo>〈</mo><mi>X</mi><mo>,</mo><mi>d</mi><mo>〉</mo></math></span>, <span><math><mo>|</mo><mi>X</mi><mo>|</mo><mo>≥</mo><msup><mrow><mn>2</mn></mrow><mrow><msub><mrow><mi>ℵ</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></msup></math></span>” in <strong>ZFA</strong>”.</div></span></li><li><span>4.</span><span><div>“For every uncountable sequentially compact metric space <span><math><mo>〈</mo><mi>X</mi><mo>,</mo><mi>d</mi><mo>〉</mo></math></span> <span><math><mo>|</mo><mi>X</mi><mo>|</mo><mo>≥</mo><msup><mrow><mn>2</mn></mrow><mrow><msub><mrow><mi>ℵ</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></msup></math></span>” does not imply “For every uncountable compact metric space <span><math><mo>〈</mo><mi>X</mi><mo>,</mo><mi>d</mi><mo>〉</mo></math></span>, <span><math><mo>|</mo><mi>X</mi><mo>|</mo><mo>=</mo><msup><mrow><mn>2</mn></mrow><mrow><msub><mrow><mi>ℵ</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></msup></math></span>” in <strong>ZFA</strong>.</div></span></li><li><span>5.</span><span><div>“For every uncountable sequentially compact metric space <span><math><mo>〈</mo><mi>X</mi><mo>,</mo><mi>d</mi><mo>","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}
Pub Date : 2025-04-08DOI: 10.1016/j.apal.2025.103601
Arturo Rodríguez Fanlo , Ori Segel
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).
{"title":"Completeness in local positive logic","authors":"Arturo Rodríguez Fanlo , 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}
Pub Date : 2025-04-07DOI: 10.1016/j.apal.2025.103602
Taras Banakh , Robert Rałowski , Szymon Żeberski
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.
{"title":"The Set-Cover game and non-measurable unions","authors":"Taras Banakh , Robert Rałowski , 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}
Pub Date : 2025-04-04DOI: 10.1016/j.apal.2025.103586
Liang Yu
We prove that, assuming ZF, and restricted to any -pointed set, Chaitin's is not injective for any universal prefix-free Turing machine U, and that fails to be degree invariant in a very strong sense, answering several recent questions in descriptive set theory. Moreover, we show that under , every function f mapping x to x-random must be uncountable-to-one over an upper cone of Turing degrees.
我们证明，假设 ZF，并限制于任何 ≤T 点集，柴廷的ΩU:x↦ΩUx=∑Ux(σ)↓2-|σ| 对于任何通用无前缀图灵机 U 都不是注入式的，并且ΩUx 在非常强的意义上不具有度不变性，这回答了描述集合论中最近的几个问题。此外，我们还证明了在 ZF+AD 下，映射 x 到 x-random 的每个函数 f 都必须在图灵度的上锥上是不可数到一的。
{"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}
Pub Date : 2025-03-25DOI: 10.1016/j.apal.2025.103585
M. Drzewiecka , A. Ivanov , B. Mokry
Let and be the closure of the conjugacy class of ρ in . We show that contains a conjugacy class, say C, which is comeagre in . We describe representatives of C. Furthermore, we show that the family of finite partial maps extendable to elements of C has the cofinal amalgamation property.
{"title":"Generics in invariant subsets of the group of order preserving permutations of Q","authors":"M. Drzewiecka , A. Ivanov , 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><</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><</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}
Pub Date : 2025-03-21DOI: 10.1016/j.apal.2025.103584
Moti Gitik , Sittinon Jirattikansakul
Continuing [1], 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.
{"title":"Extender-based Magidor-Radin forcings without top extenders","authors":"Moti Gitik , 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}
Pub Date : 2025-03-19DOI: 10.1016/j.apal.2025.103583
Victoria Gitman , Jonathan Osinski
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 κ is the upward Löwenheim-Skolem-Tarski number (ULST number) of a logic if it is the least cardinal with the property that whenever M is a model of size at least κ satisfying a sentence φ in , then there are arbitrarily large models satisfying φ and having M 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.
We define the strong 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.
{"title":"Upward Löwenheim-Skolem-Tarski numbers for abstract logics","authors":"Victoria Gitman , 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}
Pub Date : 2025-03-13DOI: 10.1016/j.apal.2025.103581
Martina Iannella , Alberto Marcone , Luca Motto Ros , Vadim Weinstein
Given a nonempty set of linear orders, we say that the linear order L is -convex embeddable into the linear order if it is possible to partition L into convex sets indexed by some element of which are isomorphic to convex subsets of ordered in the same way. This notion generalizes convex embeddability and (finite) piecewise convex embeddability (both studied in [13]), which are the special cases and . 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.
{"title":"Piecewise convex embeddability on linear orders","authors":"Martina Iannella , Alberto Marcone , Luca Motto Ros , 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}
Pub Date : 2025-03-12DOI: 10.1016/j.apal.2025.103582
Tristan van der Vlugt
We will give an overview of four families of cardinal characteristics defined on subspaces of the generalised Baire space , where κ is strongly inaccessible and . 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.
{"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}
Pub Date : 2025-03-10DOI: 10.1016/j.apal.2025.103579
Piotr Borodulin-Nadzieja , Jonathan Cancino-Manríquez , Adam Morawski
We answer in negative the problem if the existence of a P-measure implies the existence of a P-point. Namely, we show that if we add random reals to a certain ‘unique P-point’ model, then in the resulting model we will have a P-measure but not P-points. Also, we investigate the question if there is a P-measure in the Silver model. We show that rapid filters cannot be extended to a P-measure in the extension by ω product of Silver forcings and that in the model obtained by the countable support -iteration of countable product of Silver forcings there are no P-measures of countable Maharam type.
{"title":"P-measures in models without P-points","authors":"Piotr Borodulin-Nadzieja , Jonathan Cancino-Manríquez , Adam Morawski","doi":"10.1016/j.apal.2025.103579","DOIUrl":"10.1016/j.apal.2025.103579","url":null,"abstract":"<div><div>We answer in negative the problem if the existence of a P-measure implies the existence of a P-point. Namely, we show that if we add random reals to a certain ‘unique P-point’ model, then in the resulting model we will have a P-measure but not P-points. Also, we investigate the question if there is a P-measure in the Silver model. We show that rapid filters cannot be extended to a P-measure in the extension by <em>ω</em> product of Silver forcings and that in the model obtained by the countable support <span><math><msub><mrow><mi>ω</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-iteration of countable product of Silver forcings there are no P-measures of countable Maharam type.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 7","pages":"Article 103579"},"PeriodicalIF":0.6,"publicationDate":"2025-03-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143643186","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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