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Revisiting the conservativity of fixpoints over intuitionistic arithmetic 直觉算术上不动点的保守性修正
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2023-07-28 DOI: 10.1007/s00153-023-00878-2
Mattias Granberg Olsson, Graham E. Leigh
{"title":"Revisiting the conservativity of fixpoints over intuitionistic arithmetic","authors":"Mattias Granberg Olsson,&nbsp;Graham E. Leigh","doi":"10.1007/s00153-023-00878-2","DOIUrl":"10.1007/s00153-023-00878-2","url":null,"abstract":"<div><p>This paper presents a novel proof of the conservativity of the intuitionistic theory of strictly positive fixpoints, <span>(widehat{{textrm{ID}}}{}_{1}^{{textrm{i}}}{})</span>, over Heyting arithmetic (<span>({textrm{HA}})</span>), originally proved in full generality by Arai (Ann Pure Appl Log 162:807–815, 2011. https://doi.org/10.1016/j.apal.2011.03.002). The proof embeds <span>(widehat{{textrm{ID}}}{}_{1}^{{textrm{i}}}{})</span> into the corresponding theory over Beeson’s logic of partial terms and then uses two consecutive interpretations, a realizability interpretation of this theory into the subtheory generated by almost negative fixpoints, and a direct interpretation into Heyting arithmetic with partial terms using a hierarchy of satisfaction predicates for almost negative formulae. It concludes by applying van den Berg and van Slooten’s result (Indag Math 29:260–275, 2018. https://doi.org/10.1016/j.indag.2017.07.009) that Heyting arithmetic with partial terms plus the schema of self realizability for arithmetic formulae is conservative over <span>({textrm{HA}})</span>.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"63 1-2","pages":"61 - 87"},"PeriodicalIF":0.3,"publicationDate":"2023-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00153-023-00878-2.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47599322","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
Turing degrees and randomness for continuous measures 连续测度的图灵度与随机性
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2023-06-15 DOI: 10.1007/s00153-023-00873-7
Mingyang Li, Jan Reimann
{"title":"Turing degrees and randomness for continuous measures","authors":"Mingyang Li,&nbsp;Jan Reimann","doi":"10.1007/s00153-023-00873-7","DOIUrl":"10.1007/s00153-023-00873-7","url":null,"abstract":"<div><p>We study degree-theoretic properties of reals that are not random with respect to any continuous probability measure (NCR). To this end, we introduce a family of generalized Hausdorff measures based on the iterates of the “dissipation” function of a continuous measure and study the effective nullsets given by the corresponding Solovay tests. We introduce two constructions that preserve non-randomness with respect to a given continuous measure. This enables us to prove the existence of NCR reals in a number of Turing degrees. In particular, we show that every <span>(Delta ^0_2)</span>-degree contains an NCR element.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"63 1-2","pages":"39 - 59"},"PeriodicalIF":0.3,"publicationDate":"2023-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49534997","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
Models of ({{textsf{ZFA}}}) in which every linearly ordered set can be well ordered ({{textsf{ZFA}}})的模型，其中每个线性有序集都可以是有序的
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2023-06-13 DOI: 10.1007/s00153-023-00871-9
Paul Howard, Eleftherios Tachtsis
{"title":"Models of ({{textsf{ZFA}}}) in which every linearly ordered set can be well ordered","authors":"Paul Howard,&nbsp;Eleftherios Tachtsis","doi":"10.1007/s00153-023-00871-9","DOIUrl":"10.1007/s00153-023-00871-9","url":null,"abstract":"<div><p>We provide a general criterion for Fraenkel–Mostowski models of <span>({textsf{ZFA}})</span> (i.e. Zermelo–Fraenkel set theory weakened to permit the existence of atoms) which implies “every linearly ordered set can be well ordered” (<span>({textsf{LW}})</span>), and look at six models for <span>({textsf{ZFA}})</span> which satisfy this criterion (and thus <span>({textsf{LW}})</span> is true in these models) and “every Dedekind finite set is finite” (<span>({textsf{DF}}={textsf{F}})</span>) is true, and also consider various forms of choice for well-ordered families of well orderable sets in these models. In Model 1, the axiom of multiple choice for countably infinite families of countably infinite sets (<span>({textsf{MC}}_{aleph _{0}}^{aleph _{0}})</span>) is false. It was the open question of whether or not such a model exists (from Howard and Tachtsis “On metrizability and compactness of certain products without the Axiom of Choice”) that provided the motivation for this paper. In Model 2, which is constructed by first choosing an uncountable regular cardinal in the ground model, a strong form of Dependent choice is true, while the axiom of choice for well-ordered families of finite sets (<span>({textsf{AC}}^{{textsf{WO}}}_{{textsf{fin}}})</span>) is false. Also in this model the axiom of multiple choice for well-ordered families of well orderable sets fails. Model 3 is similar to Model 2 except for the status of <span>({textsf{AC}}^{{textsf{WO}}}_{{textsf{fin}}})</span> which is unknown. Models 4 and 5 are variations of Model 3. In Model 4 <span>({textsf{AC}}_{textrm{fin}}^{{textsf{WO}}})</span> is true. The construction of Model 5 begins by choosing a regular successor cardinal in the ground model. Model 6 is the only one in which <span>(2{mathfrak {m}} = {mathfrak {m}})</span> for every infinite cardinal number <span>({mathfrak {m}})</span>. We show that the union of a well-ordered family of well orderable sets is well orderable in Model 6 and that the axiom of multiple countable choice is false.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"62 7-8","pages":"1131 - 1157"},"PeriodicalIF":0.3,"publicationDate":"2023-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50023467","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
Models of ZFAdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$${{textsf{ZFA}}}$$end{document} in which every linearly ZFAdocumentclass[12pt]｛minimum｝usepackage｛amsmath｝usepackage｛wasysym｝usepackup｛amsfonts｝usecpackage{amssymb｝usecpackage{amsbsy｝usecPackage｛mathrsfs｝usepackage{upgeek｝setlength｛oddsedmargin｝｛-69pt｝ begin｛document｝$$｛textsf｛ZFA｝｝｝$｝
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2023-06-13 DOI: 10.1007/s00153-023-00871-9
Paul Howard, E. Tachtsis
{"title":"Models of ZFAdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$${{textsf{ZFA}}}$$end{document} in which every linearly ","authors":"Paul Howard, E. Tachtsis","doi":"10.1007/s00153-023-00871-9","DOIUrl":"https://doi.org/10.1007/s00153-023-00871-9","url":null,"abstract":"","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"62 1","pages":"1131 - 1157"},"PeriodicalIF":0.3,"publicationDate":"2023-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42828924","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The fixed point and the Craig interpolation properties for sublogics of (textbf{IL}) $$textbf{IL}子逻辑的不动点和Craig插值性质$$
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2023-06-10 DOI: 10.1007/s00153-023-00882-6
Sohei Iwata, Taishi Kurahashi, Yuya Okawa
{"title":"The fixed point and the Craig interpolation properties for sublogics of (textbf{IL})","authors":"Sohei Iwata,&nbsp;Taishi Kurahashi,&nbsp;Yuya Okawa","doi":"10.1007/s00153-023-00882-6","DOIUrl":"10.1007/s00153-023-00882-6","url":null,"abstract":"<div><p>We study the fixed point property and the Craig interpolation property for sublogics of the interpretability logic <span>(textbf{IL})</span>. We provide a complete description of these sublogics concerning the uniqueness of fixed points, the fixed point property and the Craig interpolation property.\u0000</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"63 1-2","pages":"1 - 37"},"PeriodicalIF":0.3,"publicationDate":"2023-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00153-023-00882-6.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43405479","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 the complexity of the theory of a computably presented metric structure 论可计算度量结构理论的复杂性
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2023-06-09 DOI: 10.1007/s00153-023-00884-4
Caleb Camrud, Isaac Goldbring, Timothy H. McNicholl
{"title":"On the complexity of the theory of a computably presented metric structure","authors":"Caleb Camrud,&nbsp;Isaac Goldbring,&nbsp;Timothy H. McNicholl","doi":"10.1007/s00153-023-00884-4","DOIUrl":"10.1007/s00153-023-00884-4","url":null,"abstract":"<div><p>We consider the complexity (in terms of the arithmetical hierarchy) of the various quantifier levels of the diagram of a computably presented metric structure. As the truth value of a sentence of continuous logic may be any real in [0, 1], we introduce two kinds of diagrams at each level: the <i>closed</i> diagram, which encapsulates weak inequalities of the form <span>(phi ^mathcal {M}le r)</span>, and the <i>open</i> diagram, which encapsulates strict inequalities of the form <span>(phi ^mathcal {M}&lt; r)</span>. We show that the closed and open <span>(Sigma _N)</span> diagrams are <span>(Pi ^0_{N+1})</span> and <span>(Sigma ^0_N)</span> respectively, and that the closed and open <span>(Pi _N)</span> diagrams are <span>(Pi ^0_N)</span> and <span>(Sigma ^0_{N + 1})</span> respectively. We then introduce effective infinitary formulas of continuous logic and extend our results to the hyperarithmetical hierarchy. Finally, we demonstrate that our results are optimal.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"62 7-8","pages":"1111 - 1129"},"PeriodicalIF":0.3,"publicationDate":"2023-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47542236","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: 2
Recursive Polish spaces 递归抛光空间
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2023-06-09 DOI: 10.1007/s00153-023-00883-5
Tyler Arant
{"title":"Recursive Polish spaces","authors":"Tyler Arant","doi":"10.1007/s00153-023-00883-5","DOIUrl":"10.1007/s00153-023-00883-5","url":null,"abstract":"<div><p>This paper is concerned with the proper way to effectivize the notion of a Polish space. A theorem is proved that shows the recursive Polish space structure is not found in the effectively open subsets of a space <span>({mathcal {X}})</span>, and we explore strong evidence that the effective structure is instead captured by the effectively open subsets of the product space <span>(mathbb {N}times {mathcal {X}})</span>.\u0000</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"62 7-8","pages":"1101 - 1110"},"PeriodicalIF":0.3,"publicationDate":"2023-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00153-023-00883-5.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44795318","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
Structure of semisimple rings in reverse and computable mathematics 半单环的结构在逆向和可计算数学中的应用
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2023-06-08 DOI: 10.1007/s00153-023-00885-3
Huishan Wu
{"title":"Structure of semisimple rings in reverse and computable mathematics","authors":"Huishan Wu","doi":"10.1007/s00153-023-00885-3","DOIUrl":"10.1007/s00153-023-00885-3","url":null,"abstract":"<div><p>This paper studies the structure of semisimple rings using techniques of reverse mathematics, where a ring is left semisimple if the left regular module is a finite direct sum of simple submodules. The structure theorem of left semisimple rings, also called Wedderburn-Artin Theorem, is a famous theorem in noncommutative algebra, says that a ring is left semisimple if and only if it is isomorphic to a finite direct product of matrix rings over division rings. We provide a proof for the theorem in <span>(mathrm RCA_{0})</span>, showing the structure theorem for computable semisimple rings. The decomposition of semisimple rings as finite direct products of matrix rings over division rings is unique. Based on an effective proof of the Jordan-Hölder Theorem for modules with composition series, we also provide an effective proof for the uniqueness of the matrix decomposition of semisimple rings in <span>(mathrm RCA_{0})</span>.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"62 7-8","pages":"1083 - 1100"},"PeriodicalIF":0.3,"publicationDate":"2023-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45835876","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
A syntactic approach to Borel functions: some extensions of Louveau’s theorem 博雷尔函数的句法方法:卢沃定理的一些扩展
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2023-06-02 DOI: 10.1007/s00153-023-00880-8
Takayuki Kihara, Kenta Sasaki
{"title":"A syntactic approach to Borel functions: some extensions of Louveau’s theorem","authors":"Takayuki Kihara,&nbsp;Kenta Sasaki","doi":"10.1007/s00153-023-00880-8","DOIUrl":"10.1007/s00153-023-00880-8","url":null,"abstract":"<div><p>Louveau showed that if a Borel set in a Polish space happens to be in a Borel Wadge class <span>(Gamma )</span>, then its <span>(Gamma )</span>-code can be obtained from its Borel code in a hyperarithmetical manner. We extend Louveau’s theorem to Borel functions: If a Borel function on a Polish space happens to be a <span>( underset{widetilde{}}{varvec{Sigma }}hbox {}_t)</span>-function, then one can find its <span>( underset{widetilde{}}{varvec{Sigma }}hbox {}_t)</span>-code hyperarithmetically relative to its Borel code. More generally, we prove extension-type, domination-type, and decomposition-type variants of Louveau’s theorem for Borel functions.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"62 7-8","pages":"1041 - 1082"},"PeriodicalIF":0.3,"publicationDate":"2023-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00153-023-00880-8.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42852550","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 the non-existence of (kappa )-mad families 关于不存在(kappa )疯狂家庭
IF 0.3 4区 数学
Archive for Mathematical Logic Pub Date : 2023-05-23 DOI: 10.1007/s00153-023-00874-6
Haim Horowitz, Saharon Shelah
{"title":"On the non-existence of (kappa )-mad families","authors":"Haim Horowitz,&nbsp;Saharon Shelah","doi":"10.1007/s00153-023-00874-6","DOIUrl":"10.1007/s00153-023-00874-6","url":null,"abstract":"<div><p>Starting from a model with a Laver-indestructible supercompact cardinal <span>(kappa )</span>, we construct a model of <span>(ZF+DC_{kappa })</span> where there are no <span>(kappa )</span>-mad families.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":"62 7-8","pages":"1033 - 1039"},"PeriodicalIF":0.3,"publicationDate":"2023-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00153-023-00874-6.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50044367","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
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