发布求助
文献互助 智能选刊 最新文献

Archive for Mathematical Logic最新文献

筛选
英文 中文
Degrees of relations on canonically ordered natural numbers and integers
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2024-10-28 DOI: 10.1007/s00153-024-00942-5
Nikolay Bazhenov, Dariusz Kalociński, Michał Wrocławski
{"title":"Degrees of relations on canonically ordered natural numbers and integers","authors":"Nikolay Bazhenov,&nbsp;Dariusz Kalociński,&nbsp;Michał Wrocławski","doi":"10.1007/s00153-024-00942-5","DOIUrl":"10.1007/s00153-024-00942-5","url":null,"abstract":"<div><p>We investigate the degree spectra of computable relations on canonically ordered natural numbers <span>((omega ,&lt;))</span> and integers <span>((zeta ,&lt;))</span>. As for <span>((omega ,&lt;))</span>, we provide several criteria that fix the degree spectrum of a computable relation to all c.e. or to all <span>(Delta _2)</span> degrees; this includes the complete characterization of the degree spectra of so-called computable block functions that have only finitely many types of blocks. Compared to Bazhenov et al. (in: LIPIcs, vol 219, pp 8:1–8:20, 2022), we obtain a more general solution to the problem regarding possible degree spectra on <span>((omega ,&lt;))</span>, answering the question whether there are infinitely many such spectra. As for <span>((zeta ,&lt;))</span>, we prove the following dichotomy result: given an arbitrary computable relation <i>R</i> on <span>((zeta ,&lt;))</span>, its degree spectrum is either trivial or it contains all c.e. degrees. This result, and the proof techniques required to solve it, extend the analogous theorem for <span>((omega ,&lt;))</span> obtained by Wright (Computability 7:349–365, 2018), and provide initial insight to Wright’s question whether such a dichotomy holds on computable ill-founded linear orders. This article is an extended version of Bazhenov et al. (in: LIPIcs, vol 219, pp 8:1–8:20, 2022).</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"64 1-2","pages":"299 - 331"},"PeriodicalIF":0.3,"publicationDate":"2024-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00153-024-00942-5.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143388706","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
A characterization of strongly computable finite factorization domains 强可计算有限因式分解域的表征
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2024-10-12 DOI: 10.1007/s00153-024-00941-6
Geraldo Soto-Rosa, Victor Ocasio-González
{"title":"A characterization of strongly computable finite factorization domains","authors":"Geraldo Soto-Rosa,&nbsp;Victor Ocasio-González","doi":"10.1007/s00153-024-00941-6","DOIUrl":"10.1007/s00153-024-00941-6","url":null,"abstract":"<div><p>In recent research, the prime and irreducible elements of strong finite factorization domains were studied. It was shown that strongly computable strong finite factorization domains (SCSFFD) have necessarily computable irreducible elements and a computable division algorithm. However, the question of how to best classify this class of structures is left unanswered. This work provides a classification for SCSFFDs by showing the existence of a computable norm where norm-form equations can be solved computably. This classification provides the intuition to extend further the notion of strong computability to finite factorization domains in general.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"64 1-2","pages":"333 - 349"},"PeriodicalIF":0.3,"publicationDate":"2024-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143388891","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
Different covering numbers of compact tree ideals 紧凑树理想的不同覆盖数
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2024-08-16 DOI: 10.1007/s00153-024-00933-6
Jelle Mathis Kuiper, Otmar Spinas
{"title":"Different covering numbers of compact tree ideals","authors":"Jelle Mathis Kuiper,&nbsp;Otmar Spinas","doi":"10.1007/s00153-024-00933-6","DOIUrl":"10.1007/s00153-024-00933-6","url":null,"abstract":"<div><p>We investigate the covering numbers of some ideals on <span>({^{omega }}{2}{})</span> associated with tree forcings. We prove that the covering of the Sacks ideal remains small in the Silver and uniform Sacks model, respectively, and that the coverings of the uniform Sacks ideal and the Mycielski ideal, <span>({mathfrak {C}_{2}})</span>, remain small in the Sacks model.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"64 1-2","pages":"259 - 278"},"PeriodicalIF":0.3,"publicationDate":"2024-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00153-024-00933-6.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142221700","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
On categoricity of scattered linear orders of constructive ranks 论构造等级的分散线性阶的分类性
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2024-08-16 DOI: 10.1007/s00153-024-00934-5
Andrey Frolov, Maxim Zubkov
{"title":"On categoricity of scattered linear orders of constructive ranks","authors":"Andrey Frolov,&nbsp;Maxim Zubkov","doi":"10.1007/s00153-024-00934-5","DOIUrl":"10.1007/s00153-024-00934-5","url":null,"abstract":"<div><p>In this article we investigate the complexity of isomorphisms between scattered linear orders of constructive ranks. We give the general upper bound and prove that this bound is sharp. Also, we construct examples showing that the categoricity level of a given scattered linear order can be an arbitrary ordinal from 3 to the upper bound, except for the case when the ordinal is the successor of a limit ordinal. The existence question of the scattered linear orders whose categoricity level equals the successor of a limit ordinal is still open.\u0000</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"64 1-2","pages":"279 - 297"},"PeriodicalIF":0.3,"publicationDate":"2024-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142221698","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
The provably total functions of basic arithmetic and its extensions 基本算术及其扩展的可证实总函数
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2024-08-14 DOI: 10.1007/s00153-024-00939-0
Mohammad Ardeshir, Erfan Khaniki, Mohsen Shahriari
{"title":"The provably total functions of basic arithmetic and its extensions","authors":"Mohammad Ardeshir,&nbsp;Erfan Khaniki,&nbsp;Mohsen Shahriari","doi":"10.1007/s00153-024-00939-0","DOIUrl":"10.1007/s00153-024-00939-0","url":null,"abstract":"<div><p>We study Basic Arithmetic, <span>(textsf{BA})</span> introduced by Ruitenburg (Notre Dame J Formal Logic 39:18–46, 1998). <span>(textsf{BA})</span> is an arithmetical theory based on basic logic which is weaker than intuitionistic logic. We show that the class of the provably total recursive functions of <span>(textsf{BA})</span> is a <i>proper</i> sub-class of the primitive recursive functions. Three extensions of <span>(textsf{BA})</span>, called <span>(textsf{BA}+mathsf U)</span>, <span>(mathsf {BA_{mathrm c}})</span> and <span>(textsf{EBA})</span> are investigated with relation to their provably total recursive functions. It is shown that the provably total recursive functions of these three extensions of <span>(textsf{BA})</span> are <i>exactly</i> the primitive recursive functions. Moreover, among other things, it is shown that the well-known MRDP theorem does not hold in <span>(textsf{BA})</span>, <span>(textsf{BA}+mathsf U)</span>, <span>(mathsf {BA_{mathrm c}})</span>, but holds in <span>(textsf{EBA})</span>.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"64 1-2","pages":"205 - 257"},"PeriodicalIF":0.3,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142221699","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
Undecidability of indecomposable polynomial rings 不可分解多项式环的不可判定性
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2024-08-12 DOI: 10.1007/s00153-024-00936-3
Marco Barone, Nicolás Caro-Montoya, Eudes Naziazeno
{"title":"Undecidability of indecomposable polynomial rings","authors":"Marco Barone,&nbsp;Nicolás Caro-Montoya,&nbsp;Eudes Naziazeno","doi":"10.1007/s00153-024-00936-3","DOIUrl":"10.1007/s00153-024-00936-3","url":null,"abstract":"<div><p>By using algebraic properties of (commutative unital) indecomposable polynomial rings we achieve results concerning their first-order theory, namely: interpretability of arithmetic and a uniform proof of undecidability of their full theory, both in the language of rings without parameters. This vastly extends the scope of a method due to <span>Raphael Robinson</span>, which deals with a restricted class of polynomial integral domains.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"64 1-2","pages":"185 - 203"},"PeriodicalIF":0.3,"publicationDate":"2024-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141940768","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
Punctually presented structures II: comparing presentations 按时展示的结构 II：展示比较
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2024-08-08 DOI: 10.1007/s00153-024-00940-7
Marina Dorzhieva, Rodney Downey, Ellen Hammatt, Alexander G. Melnikov, Keng Meng Ng
{"title":"Punctually presented structures II: comparing presentations","authors":"Marina Dorzhieva,&nbsp;Rodney Downey,&nbsp;Ellen Hammatt,&nbsp;Alexander G. Melnikov,&nbsp;Keng Meng Ng","doi":"10.1007/s00153-024-00940-7","DOIUrl":"10.1007/s00153-024-00940-7","url":null,"abstract":"<div><p>We investigate the problem of punctual (fully primitive recursive) presentability of algebraic structures up to primitive recursive and computable isomorphism. We show that for mono-unary structures and undirected graphs, if a structure is not punctually categorical then it has infinitely many punctually non-isomorphic punctual presentations. We also show that the punctual degrees of any computably almost rigid structure as well as the order (<span>(mathbb {Z},&lt;)</span>) are dense. Finally we characterise the Boolean algebras which have a punctually 1-decidable presentation that is computably isomorphic to a 1-decidable presentation.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"64 1-2","pages":"159 - 184"},"PeriodicalIF":0.3,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00153-024-00940-7.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141928836","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
On some (Sigma ^{B}_{0})-formulae generalizing counting principles over (V^{0}) 关于在 $$V^{0}$ 上概括计数原理的一些 $$Sigma ^{B}_{0}$ 公式
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2024-07-22 DOI: 10.1007/s00153-024-00938-1
Eitetsu Ken
{"title":"On some (Sigma ^{B}_{0})-formulae generalizing counting principles over (V^{0})","authors":"Eitetsu Ken","doi":"10.1007/s00153-024-00938-1","DOIUrl":"10.1007/s00153-024-00938-1","url":null,"abstract":"<div><p>We formalize various counting principles and compare their strengths over <span>(V^{0})</span>. In particular, we conjecture the following mutual independence between:</p><ul>\u0000 <li>\u0000 <p>a uniform version of modular counting principles and the pigeonhole principle for injections,</p>\u0000 </li>\u0000 <li>\u0000 <p>a version of the oddtown theorem and modular counting principles of modulus <i>p</i>, where <i>p</i> is any natural number which is not a power of 2,</p>\u0000 </li>\u0000 <li>\u0000 <p>and a version of Fisher’s inequality and modular counting principles.</p>\u0000 </li>\u0000 </ul><p> Then, we give sufficient conditions to prove them. We give a variation of the notion of <i>PHP</i>-tree and <i>k</i>-evaluation to show that any Frege proof of the pigeonhole principle for injections admitting the uniform counting principle as an axiom scheme cannot have <i>o</i>(<i>n</i>)-evaluations. As for the remaining two, we utilize well-known notions of <i>p</i>-tree and <i>k</i>-evaluation and reduce the problems to the existence of certain families of polynomials witnessing violations of the corresponding combinatorial principles with low-degree Nullstellensatz proofs from the violation of the modular counting principle in concern.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"64 1-2","pages":"117 - 158"},"PeriodicalIF":0.3,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00153-024-00938-1.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141784682","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
On absorption’s formula definable semigroups of complete theories 论完整理论的吸收式可定义半群
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2024-07-20 DOI: 10.1007/s00153-024-00937-2
Mahsut Bekenov, Aida Kassatova, Anvar Nurakunov
{"title":"On absorption’s formula definable semigroups of complete theories","authors":"Mahsut Bekenov,&nbsp;Aida Kassatova,&nbsp;Anvar Nurakunov","doi":"10.1007/s00153-024-00937-2","DOIUrl":"10.1007/s00153-024-00937-2","url":null,"abstract":"<div><p>On the set of all first-order complete theories <span>(T(sigma ))</span> of a language <span>(sigma )</span> we define a binary operation <span>({cdot })</span> by the rule: <span>(Tcdot S= {{,textrm{Th},}}({Atimes Bmid Amodels T ,,text {and},, Bmodels S}))</span> for any complete theories <span>(T, Sin T(sigma ))</span>. The structure <span>(langle T(sigma );cdot rangle )</span> forms a commutative semigroup. A subsemigroup <i>S</i> of <span>(langle T(sigma );cdot rangle )</span> is called an <i>absorption’s formula definable semigroup</i> if there is a complete theory <span>(Tin T(sigma ))</span> such that <span>(S=langle {Xin T(sigma )mid Xcdot T=T};cdot rangle )</span>. In this event we say that a theory <i>T</i> <i>absorbs</i> <i>S</i>. In the article we show that for any absorption’s formula definable semigroup <i>S</i> the class <span>({{,textrm{Mod},}}(S)={Ain {{,textrm{Mod},}}(sigma )mid Amodels T_0,,text {for some},, T_0in S})</span> is axiomatizable, and there is an idempotent element <span>(Tin S)</span> that absorbs <i>S</i>. Moreover, <span>({{,textrm{Mod},}}(S))</span> is finitely axiomatizable provided <i>T</i> is finitely axiomatizable. We also prove that <span>({{,textrm{Mod},}}(S))</span> is a quasivariety (variety) provided <i>T</i> is an universal (a positive universal) theory. Some examples are provided.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"64 1-2","pages":"107 - 116"},"PeriodicalIF":0.3,"publicationDate":"2024-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141743817","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
Intuitionistic sets and numbers: small set theory and Heyting arithmetic 直观集与数：小集理论与海廷算术
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2024-06-18 DOI: 10.1007/s00153-024-00935-4
Stewart Shapiro, Charles McCarty, Michael Rathjen
{"title":"Intuitionistic sets and numbers: small set theory and Heyting arithmetic","authors":"Stewart Shapiro,&nbsp;Charles McCarty,&nbsp;Michael Rathjen","doi":"10.1007/s00153-024-00935-4","DOIUrl":"10.1007/s00153-024-00935-4","url":null,"abstract":"<div><p>It has long been known that (classical) Peano arithmetic is, in some strong sense, “equivalent” to the variant of (classical) Zermelo–Fraenkel set theory (including choice) in which the axiom of infinity is replaced by its negation. The intended model of the latter is the set of hereditarily finite sets. The connection between the theories is so tight that they may be taken as notational variants of each other. Our purpose here is to develop and establish a constructive version of this. We present an intuitionistic theory of the hereditarily finite sets, and show that it is definitionally equivalent to Heyting Arithmetic <span>HA</span>, in a sense to be made precise. Our main target theory, the intuitionistic small set theory <span>SST</span> is remarkably simple, and intuitive. It has just one non-logical primitive, for membership, and three straightforward axioms plus one axiom scheme. We locate our theory within intuitionistic mathematics generally.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"64 1-2","pages":"79 - 105"},"PeriodicalIF":0.3,"publicationDate":"2024-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00153-024-00935-4.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141503670","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
下一页 尾页
0
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整，为了账户安全，请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
现在去查看 取消
×
提示
确定
请完成安全验证×
相关产品
×
本文献相关产品
联系我们：info@booksci.cn Book学术提供免费学术资源搜索服务，方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号：481959085
Book学术官方微信