We study the meaning of “adding a constant to a language” for any doctrine, and “adding an axiom to a theory” for a primary doctrine, by showing how these are actually two instances of the same construction. We prove their universal properties, and how these constructions are compatible with additional structure on the doctrine. Existence of Kleisli object for comonads in the 2-category of indexed poset is proved in order to build these constructions.
{"title":"Adding a constant and an axiom to a doctrine","authors":"Francesca Guffanti","doi":"10.1002/malq.202300053","DOIUrl":"https://doi.org/10.1002/malq.202300053","url":null,"abstract":"<p>We study the meaning of “adding a constant to a language” for any doctrine, and “adding an axiom to a theory” for a primary doctrine, by showing how these are actually two instances of the same construction. We prove their universal properties, and how these constructions are compatible with additional structure on the doctrine. Existence of Kleisli object for comonads in the 2-category of indexed poset is proved in order to build these constructions.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"70 3","pages":"294-332"},"PeriodicalIF":0.4,"publicationDate":"2024-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142404689","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}
<p>In this paper, we investigate the relationships between the cardinalities of the set of injections, the set of surjections, and the set of all functions on a set which is of cardinality <span></span><math> <semantics> <mi>m</mi> <annotation>$mathfrak {m}$</annotation> </semantics></math>, denoted by <span></span><math> <semantics> <mrow> <mi>I</mi> <mo>(</mo> <mi>m</mi> <mo>)</mo> </mrow> <annotation>$I(mathfrak {m})$</annotation> </semantics></math>, <span></span><math> <semantics> <mrow> <mi>J</mi> <mo>(</mo> <mi>m</mi> <mo>)</mo> </mrow> <annotation>$J(mathfrak {m})$</annotation> </semantics></math> and <span></span><math> <semantics> <msup> <mi>m</mi> <mi>m</mi> </msup> <annotation>$mathfrak {m}^mathfrak {m}$</annotation> </semantics></math>, respectively. Among our results, we show that “<span></span><math> <semantics> <mrow> <msup> <mo>seq</mo> <mrow> <mn>1</mn> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <mi>m</mi> <mo>)</mo> </mrow> <mo>≠</mo> <mi>I</mi> <mrow> <mo>(</mo> <mi>m</mi> <mo>)</mo> </mrow> <mo>≠</mo> <mo>seq</mo> <mrow> <mo>(</mo> <mi>m</mi> <mo>)</mo> </mrow> </mrow> <annotation>$operatorname{seq}^{1-1}(mathfrak {m})ne I(mathfrak {m})ne operatorname{seq}(mathfrak {m})$</annotation> </semantics></math>”, “<span></span><math> <semantics> <mrow> <msup> <mo>seq</mo> <mrow> <mn>1</mn> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <mi>m</mi> <mo>)</mo> </mrow> <mo>≠</mo> <mi>J</mi> <mrow> <mo>(</mo> <mi>m</mi> <mo>)</mo> </mrow> <mo>≠</
在本文中，我们研究了注入集合、射出集合和一个集合上所有函数的集合的心数之间的关系，这些集合的心数为 m $mathfrak {m}$，分别用 I ( m ) $I(mathfrak {m})$、J ( m ) $J(mathfrak {m})$和 m m $mathfrak {m}^mathfrak {m}$表示。Among our results, we show that “ seq 1 − 1 ( m ) ≠ I ( m ) ≠ seq ( m ) $operatorname{seq}^{1-1}(mathfrak {m})ne I(mathfrak {m})ne operatorname{seq}(mathfrak {m})$ ”, “ seq 1 − 1 ( m ) ≠ J ( m ) ≠ seq ( m ) $operatorname{seq}^{1-1}(mathfrak {m})ne J(mathfrak {m})ne operatorname{seq}(mathfrak {m})$ ” and “ seq 1 − 1 ( m ) < m m ≠ seq ( m ) $operatorname{seq}^{1-1}(mathfrak {m})<mathfrak {m}^mathfrak {m}ne operatorname{seq}(mathfrak {m})$ ” are provable for an arbitrary infinite cardinal m $mathfrak {m}$ , and these are the best possible results, in the Zermelo-Fraenkel set theory ( ZF $mathsf {ZF}$ ) without the Axiom of Choice.另外，我们还证明，存在一个无限红心 m $mathfrak {m}$，使得 S ( m ) = I ( m ) <； J ( m ) $S(mathfrak {m})=I(mathfrak {m})<J(mathfrak {m})$ 其中 S ( m ) $S(mathfrak {m})$表示一个集合上双射集合的万有性，这个集合的万有性为 m $mathfrak {m}$ 。
{"title":"The set of injections and the set of surjections on a set","authors":"Natthajak Kamkru, Nattapon Sonpanow","doi":"10.1002/malq.202300059","DOIUrl":"https://doi.org/10.1002/malq.202300059","url":null,"abstract":"<p>In this paper, we investigate the relationships between the cardinalities of the set of injections, the set of surjections, and the set of all functions on a set which is of cardinality <span></span><math>\u0000 <semantics>\u0000 <mi>m</mi>\u0000 <annotation>$mathfrak {m}$</annotation>\u0000 </semantics></math>, denoted by <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>I</mi>\u0000 <mo>(</mo>\u0000 <mi>m</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$I(mathfrak {m})$</annotation>\u0000 </semantics></math>, <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>J</mi>\u0000 <mo>(</mo>\u0000 <mi>m</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$J(mathfrak {m})$</annotation>\u0000 </semantics></math> and <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>m</mi>\u0000 <mi>m</mi>\u0000 </msup>\u0000 <annotation>$mathfrak {m}^mathfrak {m}$</annotation>\u0000 </semantics></math>, respectively. Among our results, we show that “<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mo>seq</mo>\u0000 <mrow>\u0000 <mn>1</mn>\u0000 <mo>−</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 </msup>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>m</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <mo>≠</mo>\u0000 <mi>I</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>m</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <mo>≠</mo>\u0000 <mo>seq</mo>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>m</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$operatorname{seq}^{1-1}(mathfrak {m})ne I(mathfrak {m})ne operatorname{seq}(mathfrak {m})$</annotation>\u0000 </semantics></math>”, “<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mo>seq</mo>\u0000 <mrow>\u0000 <mn>1</mn>\u0000 <mo>−</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 </msup>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>m</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <mo>≠</mo>\u0000 <mi>J</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>m</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <mo>≠</","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"70 3","pages":"275-285"},"PeriodicalIF":0.4,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142404725","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}
Layth Al-Hellawi, Rachael Alvir, Barbara F. Csima, Xinyue Xie
<p>Walker's cancellation theorem for abelian groups tells us that if <span></span><math> <semantics> <mi>A</mi> <annotation>$A$</annotation> </semantics></math> is finitely generated and <span></span><math> <semantics> <mi>G</mi> <annotation>$G$</annotation> </semantics></math> and <span></span><math> <semantics> <mi>H</mi> <annotation>$H$</annotation> </semantics></math> are such that <span></span><math> <semantics> <mrow> <mi>A</mi> <mi>⊕</mi> <mi>G</mi> <mo>≅</mo> <mi>A</mi> <mi>⊕</mi> <mi>H</mi> </mrow> <annotation>$A oplus G cong A oplus H$</annotation> </semantics></math>, then <span></span><math> <semantics> <mrow> <mi>G</mi> <mo>≅</mo> <mi>H</mi> </mrow> <annotation>$G cong H$</annotation> </semantics></math>. Deveau showed that the theorem can be effectivized, but not uniformly. In this paper, we expand on Deveau's initial analysis to show that the complexity of uniformly outputting an index of an isomorphism between <span></span><math> <semantics> <mi>G</mi> <annotation>$G$</annotation> </semantics></math> and <span></span><math> <semantics> <mi>H</mi> <annotation>$H$</annotation> </semantics></math>, given indices for <span></span><math> <semantics> <mi>A</mi> <annotation>$A$</annotation> </semantics></math>, <span></span><math> <semantics> <mi>G</mi> <annotation>$G$</annotation> </semantics></math>, <span></span><math> <semantics> <mi>H</mi> <annotation>$H$</annotation> </semantics></math>, the isomorphism between <span></span><math> <semantics> <mrow> <mi>A</mi> <mi>⊕</mi> <mi>G</mi> </mrow> <annotation>$A oplus G$</annotation> </semantics></math> and <span></span><math> <semantics> <mrow> <mi>A</mi> <mi>⊕</mi> <mi>H</mi> </mrow> <annotation>$A oplus H$</annotation> </semantics></math>, and the rank of <span></span><math> <semantics>
{"title":"Effectiveness of Walker's cancellation theorem","authors":"Layth Al-Hellawi, Rachael Alvir, Barbara F. Csima, Xinyue Xie","doi":"10.1002/malq.202400030","DOIUrl":"10.1002/malq.202400030","url":null,"abstract":"<p>Walker's cancellation theorem for abelian groups tells us that if <span></span><math>\u0000 <semantics>\u0000 <mi>A</mi>\u0000 <annotation>$A$</annotation>\u0000 </semantics></math> is finitely generated and <span></span><math>\u0000 <semantics>\u0000 <mi>G</mi>\u0000 <annotation>$G$</annotation>\u0000 </semantics></math> and <span></span><math>\u0000 <semantics>\u0000 <mi>H</mi>\u0000 <annotation>$H$</annotation>\u0000 </semantics></math> are such that <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>A</mi>\u0000 <mi>⊕</mi>\u0000 <mi>G</mi>\u0000 <mo>≅</mo>\u0000 <mi>A</mi>\u0000 <mi>⊕</mi>\u0000 <mi>H</mi>\u0000 </mrow>\u0000 <annotation>$A oplus G cong A oplus H$</annotation>\u0000 </semantics></math>, then <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 <mo>≅</mo>\u0000 <mi>H</mi>\u0000 </mrow>\u0000 <annotation>$G cong H$</annotation>\u0000 </semantics></math>. Deveau showed that the theorem can be effectivized, but not uniformly. In this paper, we expand on Deveau's initial analysis to show that the complexity of uniformly outputting an index of an isomorphism between <span></span><math>\u0000 <semantics>\u0000 <mi>G</mi>\u0000 <annotation>$G$</annotation>\u0000 </semantics></math> and <span></span><math>\u0000 <semantics>\u0000 <mi>H</mi>\u0000 <annotation>$H$</annotation>\u0000 </semantics></math>, given indices for <span></span><math>\u0000 <semantics>\u0000 <mi>A</mi>\u0000 <annotation>$A$</annotation>\u0000 </semantics></math>, <span></span><math>\u0000 <semantics>\u0000 <mi>G</mi>\u0000 <annotation>$G$</annotation>\u0000 </semantics></math>, <span></span><math>\u0000 <semantics>\u0000 <mi>H</mi>\u0000 <annotation>$H$</annotation>\u0000 </semantics></math>, the isomorphism between <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>A</mi>\u0000 <mi>⊕</mi>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation>$A oplus G$</annotation>\u0000 </semantics></math> and <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>A</mi>\u0000 <mi>⊕</mi>\u0000 <mi>H</mi>\u0000 </mrow>\u0000 <annotation>$A oplus H$</annotation>\u0000 </semantics></math>, and the rank of <span></span><math>\u0000 <semantics>\u0000 ","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"70 3","pages":"347-355"},"PeriodicalIF":0.4,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/malq.202400030","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254844","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}
Given a scale (in the sense of Shelah's pcf theory), we list various conditions ensuring that a given point is good for the scale.
给定一个标度（根据谢拉的 pcf 理论），我们列出各种条件，确保给定的点对标度有利。
{"title":"Good points for scales (and more)","authors":"Pierre Matet","doi":"10.1002/malq.202300034","DOIUrl":"10.1002/malq.202300034","url":null,"abstract":"<p>Given a scale (in the sense of Shelah's pcf theory), we list various conditions ensuring that a given point is good for the scale.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"70 3","pages":"333-346"},"PeriodicalIF":0.4,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142182741","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}
The numbers of corollaries and propositions in the proof of Theorem 3.8 on p. 260 in the article Implications of Ramsey Choice principles in by Lorenz Halbeisen, Riccardo Plati, and Saharon Shelah (doi: 10.1002/malq.202300024) are incorrect. The correct numbers are given here:
{"title":"Editorial correction for L. Halbeisen, R. Plati, and Saharon Shelah, “Implications of Ramsey Choice principles in \u0000 \u0000 ZF\u0000 $mathsf {ZF}$\u0000 ”, https://doi.org/10.1002/malq.202300024","authors":"","doi":"10.1002/malq.202430002","DOIUrl":"10.1002/malq.202430002","url":null,"abstract":"<p>The numbers of corollaries and propositions in the proof of Theorem 3.8 on p. 260 in the article <i>Implications of Ramsey Choice principles in</i> <span></span><math>\u0000 <semantics>\u0000 <mi>ZF</mi>\u0000 <annotation>$mathsf {ZF}$</annotation>\u0000 </semantics></math> by Lorenz Halbeisen, Riccardo Plati, and Saharon Shelah (doi: 10.1002/malq.202300024) are incorrect. The correct numbers are given here:</p><p>August 2024</p><p>The MLQ Editorial Office</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"70 3","pages":"356"},"PeriodicalIF":0.4,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/malq.202430002","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142182740","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}
<p>We state conditions for which a definable local homomorphism between two locally definable groups <span></span><math> <semantics> <mi>G</mi> <annotation>$mathcal {G}$</annotation> </semantics></math>, <span></span><math> <semantics> <msup> <mi>G</mi> <mo>′</mo> </msup> <annotation>$mathcal {G^{prime }}$</annotation> </semantics></math> can be uniquely extended when <span></span><math> <semantics> <mi>G</mi> <annotation>$mathcal {G}$</annotation> </semantics></math> is simply connected (Theorem 2.1). As an application of this result we obtain an easy proof of [3, Theorem 9.1] (cf. Corollary 2.3). We also prove that [3, Theorem 10.2] also holds for any definably connected definably compact semialgebraic group <span></span><math> <semantics> <mi>G</mi> <annotation>$G$</annotation> </semantics></math> not necessarily abelian over a sufficiently saturated real closed field <span></span><math> <semantics> <mi>R</mi> <annotation>$R$</annotation> </semantics></math>; namely, that the o-minimal universal covering group <span></span><math> <semantics> <mover> <mi>G</mi> <mo>∼</mo> </mover> <annotation>$widetilde{G}$</annotation> </semantics></math> of <span></span><math> <semantics> <mi>G</mi> <annotation>$G$</annotation> </semantics></math> is an open locally definable subgroup of <span></span><math> <semantics> <mover> <mrow> <mi>H</mi> <msup> <mrow> <mo>(</mo> <mi>R</mi> <mo>)</mo> </mrow> <mn>0</mn> </msup> </mrow> <mo>∼</mo> </mover> <annotation>$widetilde{H(R)^{0}}$</annotation> </semantics></math> for some <span></span><math> <semantics> <mi>R</mi> <annotation>$R$</annotation> </semantics></math>-algebraic group <span></span><math> <semantics> <mi>H</mi> <annotation>$H$</annotation> </semantics></math> (Theorem 3.3). Finally, for an abelian definably connected semialgebraic group <span></span><math> <semantics> <mi>G</mi>
{"title":"Extensions of definable local homomorphisms in o-minimal structures and semialgebraic groups","authors":"Eliana Barriga","doi":"10.1002/malq.202300028","DOIUrl":"10.1002/malq.202300028","url":null,"abstract":"<p>We state conditions for which a definable local homomorphism between two locally definable groups <span></span><math>\u0000 <semantics>\u0000 <mi>G</mi>\u0000 <annotation>$mathcal {G}$</annotation>\u0000 </semantics></math>, <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>G</mi>\u0000 <mo>′</mo>\u0000 </msup>\u0000 <annotation>$mathcal {G^{prime }}$</annotation>\u0000 </semantics></math> can be uniquely extended when <span></span><math>\u0000 <semantics>\u0000 <mi>G</mi>\u0000 <annotation>$mathcal {G}$</annotation>\u0000 </semantics></math> is simply connected (Theorem 2.1). As an application of this result we obtain an easy proof of [3, Theorem 9.1] (cf. Corollary 2.3). We also prove that [3, Theorem 10.2] also holds for any definably connected definably compact semialgebraic group <span></span><math>\u0000 <semantics>\u0000 <mi>G</mi>\u0000 <annotation>$G$</annotation>\u0000 </semantics></math> not necessarily abelian over a sufficiently saturated real closed field <span></span><math>\u0000 <semantics>\u0000 <mi>R</mi>\u0000 <annotation>$R$</annotation>\u0000 </semantics></math>; namely, that the o-minimal universal covering group <span></span><math>\u0000 <semantics>\u0000 <mover>\u0000 <mi>G</mi>\u0000 <mo>∼</mo>\u0000 </mover>\u0000 <annotation>$widetilde{G}$</annotation>\u0000 </semantics></math> of <span></span><math>\u0000 <semantics>\u0000 <mi>G</mi>\u0000 <annotation>$G$</annotation>\u0000 </semantics></math> is an open locally definable subgroup of <span></span><math>\u0000 <semantics>\u0000 <mover>\u0000 <mrow>\u0000 <mi>H</mi>\u0000 <msup>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>R</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <mn>0</mn>\u0000 </msup>\u0000 </mrow>\u0000 <mo>∼</mo>\u0000 </mover>\u0000 <annotation>$widetilde{H(R)^{0}}$</annotation>\u0000 </semantics></math> for some <span></span><math>\u0000 <semantics>\u0000 <mi>R</mi>\u0000 <annotation>$R$</annotation>\u0000 </semantics></math>-algebraic group <span></span><math>\u0000 <semantics>\u0000 <mi>H</mi>\u0000 <annotation>$H$</annotation>\u0000 </semantics></math> (Theorem 3.3). Finally, for an abelian definably connected semialgebraic group <span></span><math>\u0000 <semantics>\u0000 <mi>G</mi>\u0000 ","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"70 3","pages":"267-274"},"PeriodicalIF":0.4,"publicationDate":"2024-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/malq.202300028","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141745481","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}
We expand the classic result that is equivalent to the statement “For all , ” by proving the equivalence of many more related statements. Then, we introduce the Hartogs–Lindenbaum spectrum of a model of , and inspect the structure of these spectra in models that are obtained by a symmetric extension of a model of . We prove that all such spectra fall into a very rigid pattern.
{"title":"The Hartogs–Lindenbaum spectrum of symmetric extensions","authors":"Calliope Ryan-Smith","doi":"10.1002/malq.202300047","DOIUrl":"10.1002/malq.202300047","url":null,"abstract":"<p>We expand the classic result that <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>AC</mi>\u0000 <mi>WO</mi>\u0000 </msub>\u0000 <annotation>$mathsf {AC}_mathsf {WO}$</annotation>\u0000 </semantics></math> is equivalent to the statement “For all <span></span><math>\u0000 <semantics>\u0000 <mi>X</mi>\u0000 <annotation>$X$</annotation>\u0000 </semantics></math>, <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>ℵ</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>X</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <mo>=</mo>\u0000 <msup>\u0000 <mi>ℵ</mi>\u0000 <mo>∗</mo>\u0000 </msup>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>X</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$aleph (X)=aleph ^*(X)$</annotation>\u0000 </semantics></math>” by proving the equivalence of many more related statements. Then, we introduce the Hartogs–Lindenbaum spectrum of a model of <span></span><math>\u0000 <semantics>\u0000 <mi>ZF</mi>\u0000 <annotation>$mathsf {ZF}$</annotation>\u0000 </semantics></math>, and inspect the structure of these spectra in models that are obtained by a symmetric extension of a model of <span></span><math>\u0000 <semantics>\u0000 <mi>ZFC</mi>\u0000 <annotation>$mathsf {ZFC}$</annotation>\u0000 </semantics></math>. We prove that all such spectra fall into a very rigid pattern.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"70 2","pages":"210-223"},"PeriodicalIF":0.4,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/malq.202300047","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141720944","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}
<p>We prove that for every ultrafilter <span></span><math> <semantics> <mi>U</mi> <annotation>$mathcal {U}$</annotation> </semantics></math> on <span></span><math> <semantics> <mi>ω</mi> <annotation>$omega$</annotation> </semantics></math> there exists a filter <span></span><math> <semantics> <mi>F</mi> <annotation>$mathcal {F}$</annotation> </semantics></math> on <span></span><math> <semantics> <msup> <mn>2</mn> <mrow> <mo><</mo> <mi>ω</mi> </mrow> </msup> <annotation>$2^{<omega }$</annotation> </semantics></math> which is <span></span><math> <semantics> <mi>U</mi> <annotation>$mathcal {U}$</annotation> </semantics></math>-Menger and <span></span><math> <semantics> <mrow> <mi>χ</mi> <mo>(</mo> <mi>F</mi> <mo>)</mo> <mo>=</mo> <mi>b</mi> <mo>(</mo> <mi>U</mi> <mo>)</mo> </mrow> <annotation>$chi (mathcal {F})=mathfrak {b}(mathcal {U})$</annotation> </semantics></math>. We show that in the Cohen model there exists such <span></span><math> <semantics> <mi>F</mi> <annotation>$mathcal {F}$</annotation> </semantics></math> which are tall by using a construction of Nyikos's [10]. These answer a question of Das [2, Problem 7]. We prove that there is a Menger filter of character <span></span><math> <semantics> <mi>d</mi> <annotation>$mathfrak {d}$</annotation> </semantics></math> that is not Hurewicz in the <span></span><math> <semantics> <mi>κ</mi> <annotation>$kappa$</annotation> </semantics></math>-Cohen model where <span></span><math> <semantics> <mrow> <mi>κ</mi> <mo>></mo> <msub> <mi>ω</mi> <mn>1</mn> </msub> </mrow> <annotation>$kappa >omega _{1}$</annotation> </semantics></math> is uncountable regular. This shows that the positive answer to a question of Hernández-Gutiérrez and Szeptycki [3, Question 2.8] is consistent with <span></span><math> <semantics> <mrow> <mi>b</mi> <mo><</mo> <mi>d</mi> </mrow> <annotation>$mathfrak {b}<mathfrak {d}$</annotation> </se
{"title":"Filter-Menger set of reals in Cohen extensions","authors":"Hang Zhang, Shuguo Zhang","doi":"10.1002/malq.202300008","DOIUrl":"10.1002/malq.202300008","url":null,"abstract":"<p>We prove that for every ultrafilter <span></span><math>\u0000 <semantics>\u0000 <mi>U</mi>\u0000 <annotation>$mathcal {U}$</annotation>\u0000 </semantics></math> on <span></span><math>\u0000 <semantics>\u0000 <mi>ω</mi>\u0000 <annotation>$omega$</annotation>\u0000 </semantics></math> there exists a filter <span></span><math>\u0000 <semantics>\u0000 <mi>F</mi>\u0000 <annotation>$mathcal {F}$</annotation>\u0000 </semantics></math> on <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mn>2</mn>\u0000 <mrow>\u0000 <mo><</mo>\u0000 <mi>ω</mi>\u0000 </mrow>\u0000 </msup>\u0000 <annotation>$2^{&lt;omega }$</annotation>\u0000 </semantics></math> which is <span></span><math>\u0000 <semantics>\u0000 <mi>U</mi>\u0000 <annotation>$mathcal {U}$</annotation>\u0000 </semantics></math>-Menger and <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>χ</mi>\u0000 <mo>(</mo>\u0000 <mi>F</mi>\u0000 <mo>)</mo>\u0000 <mo>=</mo>\u0000 <mi>b</mi>\u0000 <mo>(</mo>\u0000 <mi>U</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$chi (mathcal {F})=mathfrak {b}(mathcal {U})$</annotation>\u0000 </semantics></math>. We show that in the Cohen model there exists such <span></span><math>\u0000 <semantics>\u0000 <mi>F</mi>\u0000 <annotation>$mathcal {F}$</annotation>\u0000 </semantics></math> which are tall by using a construction of Nyikos's [10]. These answer a question of Das [2, Problem 7]. We prove that there is a Menger filter of character <span></span><math>\u0000 <semantics>\u0000 <mi>d</mi>\u0000 <annotation>$mathfrak {d}$</annotation>\u0000 </semantics></math> that is not Hurewicz in the <span></span><math>\u0000 <semantics>\u0000 <mi>κ</mi>\u0000 <annotation>$kappa$</annotation>\u0000 </semantics></math>-Cohen model where <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>κ</mi>\u0000 <mo>></mo>\u0000 <msub>\u0000 <mi>ω</mi>\u0000 <mn>1</mn>\u0000 </msub>\u0000 </mrow>\u0000 <annotation>$kappa &gt;omega _{1}$</annotation>\u0000 </semantics></math> is uncountable regular. This shows that the positive answer to a question of Hernández-Gutiérrez and Szeptycki [3, Question 2.8] is consistent with <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>b</mi>\u0000 <mo><</mo>\u0000 <mi>d</mi>\u0000 </mrow>\u0000 <annotation>$mathfrak {b}&lt;mathfrak {d}$</annotation>\u0000 </se","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"70 2","pages":"224-232"},"PeriodicalIF":0.4,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141645034","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}
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