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The second-order version of Morley’s theorem on the number of countable models does not require large cardinals 莫雷可数模型数定理的二阶版本不需要大的心形数
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2024-02-14 DOI: 10.1007/s00153-024-00907-8
Franklin D. Tall, Jing Zhang
{"title":"The second-order version of Morley’s theorem on the number of countable models does not require large cardinals","authors":"Franklin D. Tall,&nbsp;Jing Zhang","doi":"10.1007/s00153-024-00907-8","DOIUrl":"10.1007/s00153-024-00907-8","url":null,"abstract":"<div><p>The consistency of a second-order version of Morley’s Theorem on the number of countable models was proved in [EHMT23] with the aid of large cardinals. We here dispense with them.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"63 3-4","pages":""},"PeriodicalIF":0.3,"publicationDate":"2024-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139764927","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}
引用次数: 0
Indestructibility and the linearity of the Mitchell ordering 坚不可摧和米切尔排序的直线性
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2024-02-13 DOI: 10.1007/s00153-024-00908-7
Arthur W. Apter
{"title":"Indestructibility and the linearity of the Mitchell ordering","authors":"Arthur W. Apter","doi":"10.1007/s00153-024-00908-7","DOIUrl":"10.1007/s00153-024-00908-7","url":null,"abstract":"<div><p>Suppose that <span>(kappa )</span> is indestructibly supercompact and there is a measurable cardinal <span>(lambda &gt; kappa )</span>. It then follows that <span>(A_0 = {delta &lt; kappa mid delta )</span> is a measurable cardinal and the Mitchell ordering of normal measures over <span>(delta )</span> is nonlinear<span>(})</span> is unbounded in <span>(kappa )</span>. If the Mitchell ordering of normal measures over <span>(lambda )</span> is also linear, then by reflection (and without any use of indestructibility), <span>(A_1= {delta &lt; kappa mid delta )</span> is a measurable cardinal and the Mitchell ordering of normal measures over <span>(delta )</span> is linear<span>(})</span> is unbounded in <span>(kappa )</span> as well. The large cardinal hypothesis on <span>(lambda )</span> is necessary. We demonstrate this by constructing via forcing two models in which <span>(kappa )</span> is supercompact and <span>(kappa )</span> exhibits an indestructibility property slightly weaker than full indestructibility but sufficient to infer that <span>(A_0)</span> is unbounded in <span>(kappa )</span> if <span>(lambda &gt; kappa )</span> is measurable. In one of these models, for every measurable cardinal <span>(delta )</span>, the Mitchell ordering of normal measures over <span>(delta )</span> is linear. In the other of these models, for every measurable cardinal <span>(delta )</span>, the Mitchell ordering of normal measures over <span>(delta )</span> is nonlinear.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"63 3-4","pages":""},"PeriodicalIF":0.3,"publicationDate":"2024-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139764739","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}
引用次数: 0
Regressive versions of Hindman’s theorem 欣德曼定理的回归版本
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2024-01-31 DOI: 10.1007/s00153-023-00901-6
Lorenzo Carlucci, Leonardo Mainardi
{"title":"Regressive versions of Hindman’s theorem","authors":"Lorenzo Carlucci,&nbsp;Leonardo Mainardi","doi":"10.1007/s00153-023-00901-6","DOIUrl":"10.1007/s00153-023-00901-6","url":null,"abstract":"<div><p>When the Canonical Ramsey’s Theorem by Erdős and Rado is applied to regressive functions, one obtains the Regressive Ramsey’s Theorem by Kanamori and McAloon. Taylor proved a “canonical” version of Hindman’s Theorem, analogous to the Canonical Ramsey’s Theorem. We introduce the restriction of Taylor’s Canonical Hindman’s Theorem to a subclass of the regressive functions, the <span>(lambda )</span>-regressive functions, relative to an adequate version of min-homogeneity and prove some results about the Reverse Mathematics of this Regressive Hindman’s Theorem and of natural restrictions of it. In particular we prove that the first non-trivial restriction of the principle is equivalent to Arithmetical Comprehension. We furthermore prove that the well-ordering-preservation principle for base-<span>(omega )</span> exponentiation is reducible to this same principle by a uniform computable reduction.\u0000</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"63 3-4","pages":""},"PeriodicalIF":0.3,"publicationDate":"2024-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00153-023-00901-6.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139647497","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}
引用次数: 0
Cut elimination for coherent theories in negation normal form 否定正则表达式中相干理论的切分消除
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2024-01-24 DOI: 10.1007/s00153-023-00902-5
Paolo Maffezioli
{"title":"Cut elimination for coherent theories in negation normal form","authors":"Paolo Maffezioli","doi":"10.1007/s00153-023-00902-5","DOIUrl":"10.1007/s00153-023-00902-5","url":null,"abstract":"<div><p>We present a cut-free sequent calculus for a class of first-order theories in negation normal form which include coherent and co-coherent theories alike. All structural rules, including cut, are admissible.\u0000</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"63 3-4","pages":""},"PeriodicalIF":0.3,"publicationDate":"2024-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00153-023-00902-5.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139553648","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}
引用次数: 0
L-domains as locally continuous sequent calculi 作为局部连续序列计算的 L 域
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2024-01-23 DOI: 10.1007/s00153-023-00903-4
Longchun Wang, Qingguo Li
{"title":"L-domains as locally continuous sequent calculi","authors":"Longchun Wang,&nbsp;Qingguo Li","doi":"10.1007/s00153-023-00903-4","DOIUrl":"10.1007/s00153-023-00903-4","url":null,"abstract":"<div><p>Inspired by a framework of multi lingual sequent calculus, we introduce a formal logical system called locally continuous sequent calculus to represent <i>L</i>-domains. By considering the logic states defined on locally continuous sequent calculi, we show that the collection of all logic states of a locally continuous sequent calculus with respect to set inclusion forms an <i>L</i>-domain, and every <i>L</i>-domain can be obtained in this way. Moreover, we define conjunctive consequence relations as morphisms between our sequent calculi, and prove that the category of locally continuous sequent calculi and conjunctive consequence relations is equivalent to that of <i>L</i>-domains and Scott-continuous functions. This result extends Abramsky’s “Domain theory in logical form” to a continuous setting.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"63 3-4","pages":""},"PeriodicalIF":0.3,"publicationDate":"2024-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139553643","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}
引用次数: 0
Prenex normalization and the hierarchical classification of formulas Prenex 标准化和公式的分层分类
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2023-12-23 DOI: 10.1007/s00153-023-00899-x
Makoto Fujiwara, Taishi Kurahashi
{"title":"Prenex normalization and the hierarchical classification of formulas","authors":"Makoto Fujiwara,&nbsp;Taishi Kurahashi","doi":"10.1007/s00153-023-00899-x","DOIUrl":"10.1007/s00153-023-00899-x","url":null,"abstract":"<div><p>Akama et al. [1] introduced a hierarchical classification of first-order formulas for a hierarchical prenex normal form theorem in semi-classical arithmetic. In this paper, we give a justification for the hierarchical classification in a general context of first-order theories. To this end, we first formalize the standard transformation procedure for prenex normalization. Then we show that the classes <span>(textrm{E}_k)</span> and <span>(textrm{U}_k)</span> introduced in [1] are exactly the classes induced by <span>(Sigma _k)</span> and <span>(Pi _k)</span> respectively via the transformation procedure in any first-order theory.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"63 3-4","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139020330","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}
引用次数: 0
Weak essentially undecidable theories of concatenation, part II 本质上不可判定的弱串联理论，第二部分
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2023-11-02 DOI: 10.1007/s00153-023-00898-y
Juvenal Murwanashyaka
{"title":"Weak essentially undecidable theories of concatenation, part II","authors":"Juvenal Murwanashyaka","doi":"10.1007/s00153-023-00898-y","DOIUrl":"10.1007/s00153-023-00898-y","url":null,"abstract":"<div><p>We show that we can interpret concatenation theories in arithmetical theories without coding sequences by identifying binary strings with <span>(2times 2)</span> matrices with determinant 1.\u0000</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"63 3-4","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00153-023-00898-y.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135933800","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}
引用次数: 0
Maximal Tukey types, P-ideals and the weak Rudin–Keisler order 最大图基类型、P-理想和弱鲁丁-凯斯勒阶
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2023-10-31 DOI: 10.1007/s00153-023-00897-z
Konstantinos A. Beros, Paul B. Larson
{"title":"Maximal Tukey types, P-ideals and the weak Rudin–Keisler order","authors":"Konstantinos A. Beros,&nbsp;Paul B. Larson","doi":"10.1007/s00153-023-00897-z","DOIUrl":"10.1007/s00153-023-00897-z","url":null,"abstract":"<div><p>In this paper, we study some new examples of ideals on <span>(omega )</span> with maximal Tukey type (that is, maximal among partial orders of size continuum). This discussion segues into an examination of a refinement of the Tukey order—known as the <i>weak Rudin–Keisler order</i>—and its structure when restricted to these ideals of maximal Tukey type. Mirroring a result of Fremlin (Note Mat 11:177–214, 1991) on the Tukey order, we also show that there is an analytic P-ideal above all other analytic P-ideals in the weak Rudin–Keisler order.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"63 3-4","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135765826","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}
引用次数: 0
On Harrop disjunction property in intermediate predicate logics 论中间谓词逻辑中的哈洛普析取属性
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2023-10-27 DOI: 10.1007/s00153-023-00895-1
Katsumasa Ishii
{"title":"On Harrop disjunction property in intermediate predicate logics","authors":"Katsumasa Ishii","doi":"10.1007/s00153-023-00895-1","DOIUrl":"10.1007/s00153-023-00895-1","url":null,"abstract":"<div><p>A partial solution to Ono’s problem P54 is given. Here Ono’s problem P54 is whether Harrop disjunction property is equivalent to disjunction property or not in intermediate predicate logics. As an application of this result it is shown that some intermediate predicate logics satisfy Harrop disjunction property.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"63 3-4","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136234132","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}
引用次数: 0
Stably embedded submodels of Henselian valued fields 汉塞尔有价域的稳定嵌入子模型
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2023-10-20 DOI: 10.1007/s00153-023-00894-2
Pierre Touchard
{"title":"Stably embedded submodels of Henselian valued fields","authors":"Pierre Touchard","doi":"10.1007/s00153-023-00894-2","DOIUrl":"10.1007/s00153-023-00894-2","url":null,"abstract":"<div><p>We show a transfer principle for the property that all types realised in a given elementary extension are definable. It can be written as follows: a Henselian valued field is stably embedded in an elementary extension if and only if its value group is stably embedded in its corresponding extension, its residue field is stably embedded in its corresponding extension, and the extension of valued fields satisfies a certain algebraic condition. We show for instance that all types over the Hahn field <span>(mathbb {R}((mathbb {Z})))</span> are definable. Similarly, all types over the quotient field of the Witt ring <span>(W(mathbb {F}_p^{text {alg}}))</span> are definable. This extends a work of Cubides and Delon and of Cubides and Ye.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"63 3-4","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135513929","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}
引用次数: 0
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