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Some model theory of quadratic geometries 二次几何的一些模型理论
arXiv - MATH - Logic Pub Date : 2024-08-19 DOI: arxiv-2408.10196
Charlotte Kestner, Nicholas Ramsey
{"title":"Some model theory of quadratic geometries","authors":"Charlotte Kestner, Nicholas Ramsey","doi":"arxiv-2408.10196","DOIUrl":"https://doi.org/arxiv-2408.10196","url":null,"abstract":"Orthogonal spaces are vector spaces together with a quadratic form whose\u0000associated bilinear form is non-degenerate. Over fields of characteristic two,\u0000there are many quadratic forms associated to a given bilinear form and\u0000quadratic geometries are structures that encode a vector space over a field of\u0000characteristic 2 with a non-degenerate bilinear form together with a space of\u0000associated quadratic forms. These structures over finite fields of\u0000characteristic 2 form an important part of the basic geometries that appear in\u0000the Lie coordinatizable structures of Cherlin and Hrushovski. We (a) describe\u0000the respective model companions of the theory of orthogonal spaces and the\u0000theory of quadratic geometries and (b) classify the pseudo-finite completions\u0000of these theories. We also (c) give a neostability-theoretic classification of\u0000the model companions and these pseudo-finite completions. This is a small step\u0000towards understanding the analogue of the Cherlin-Hrushovski theory of Lie\u0000coordinatizable structures in a setting where the involved fields may be\u0000pseudo-finite.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142188028","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Geometric theories for real number algebra without sign test or dependent choice axiom 无符号检验或从属选择公理的实数代数几何理论
arXiv - MATH - Logic Pub Date : 2024-08-19 DOI: arxiv-2408.10290
Henri Lombardi, Assia Mahboubi
{"title":"Geometric theories for real number algebra without sign test or dependent choice axiom","authors":"Henri Lombardi, Assia Mahboubi","doi":"arxiv-2408.10290","DOIUrl":"https://doi.org/arxiv-2408.10290","url":null,"abstract":"In this memoir, we seek to construct a constructive theory that is as\u0000complete as possible to describe the algebraic properties of the real number\u0000field in constructive mathematics without a dependent choice axiom. To this\u0000purpose, we use a dynamical version of geometric theories. We obtain a nice\u0000description of the algebraic properties of the real number field, but also a\u0000first outline for a constructive theory of certain o-minimal structures. The\u0000memoir we present here is an unfinished development of the article by the\u0000authors https://inria.hal.science/hal-01426164. Compared to that paper,\u0000however, we have modified the definition of continuous semialgebraic functions,\u0000in the same spirit in which Bishop defines a continuous real function as a\u0000uniformly continuous function on any bounded interval. Despite its unfinished\u0000nature and the many questions that we do not currently know how to answer, we\u0000hope that this paper will arouse interest for its original approach to the\u0000subject. This paper is an English translation of a French version on\u0000arXiv:2406.15218","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187783","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Clarifying ordinals 澄清序数
arXiv - MATH - Logic Pub Date : 2024-08-19 DOI: arxiv-2408.10367
Noah Schweber
{"title":"Clarifying ordinals","authors":"Noah Schweber","doi":"arxiv-2408.10367","DOIUrl":"https://doi.org/arxiv-2408.10367","url":null,"abstract":"We use forcing over admissible sets to show that, for every ordinal $alpha$\u0000in a club $Csubsetomega_1$, there are copies of $alpha$ such that the\u0000isomorphism between them is not computable in the join of the complete\u0000$Pi^1_1$ set relative to each copy separately. Assuming $mathsf{V=L}$, this\u0000is close to optimal; on the other hand, assuming large cardinals the same (and\u0000more) holds for every projective functional.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187782","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The breadth of constructibility degrees and definable Sierpiński's coverings 可构造度的广度和可定义的西尔皮斯基覆盖层
arXiv - MATH - Logic Pub Date : 2024-08-19 DOI: arxiv-2408.10182
Alessandro Andretta, Lorenzo Notaro
{"title":"The breadth of constructibility degrees and definable Sierpiński's coverings","authors":"Alessandro Andretta, Lorenzo Notaro","doi":"arxiv-2408.10182","DOIUrl":"https://doi.org/arxiv-2408.10182","url":null,"abstract":"Generalizing a result of T\"ornquist and Weiss, we study the connection\u0000between the existence of $varSigma_2^1$ Sierpi'{n}ski's coverings of\u0000$mathbb{R}^n$, and a cardinal invariant of the upper semi-lattice of\u0000constructibility degrees known as breadth.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187784","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Proper classes of maximal $θ$-independent families from large cardinals 来自大红心的最大θ$独立族的适当类
arXiv - MATH - Logic Pub Date : 2024-08-19 DOI: arxiv-2408.10137
Calliope Ryan-Smith
{"title":"Proper classes of maximal $θ$-independent families from large cardinals","authors":"Calliope Ryan-Smith","doi":"arxiv-2408.10137","DOIUrl":"https://doi.org/arxiv-2408.10137","url":null,"abstract":"While maximal independent families can be constructed from ZFC via Zorn's\u0000lemma, the presence of a maximal $sigma$-independent family already gives an\u0000inner model with a measurable cardinal, and Kunen has shown that from a\u0000measurable cardinal one can construct a forcing extension in which there is a\u0000maximal $sigma$-independent family. We extend this technique to construct\u0000proper classes of maximal $theta$-independent families for various uncountable\u0000$theta$. In the first instance, a single $theta^+$-strongly compact cardinal\u0000has a set-generic extension with a proper class of maximal $theta$-independent\u0000families. In the second, we take a class-generic extension of a model with a\u0000proper class of measurable cardinals to obtain a proper class of $theta$ for\u0000which there is a maximal $theta$-independent family.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187785","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
An Introduction to Categorical Proof Theory 分类证明理论导论
arXiv - MATH - Logic Pub Date : 2024-08-18 DOI: arxiv-2408.09488
Amirhossein Akbar Tabatabai
{"title":"An Introduction to Categorical Proof Theory","authors":"Amirhossein Akbar Tabatabai","doi":"arxiv-2408.09488","DOIUrl":"https://doi.org/arxiv-2408.09488","url":null,"abstract":"These expanded lecture notes are based on a tutorial on categorical proof\u0000theory presented at the summer school associated with the conference \"Topology,\u0000Algebra, and Categories in Logic 2021-2022.\" The chapter delves into various\u0000applications of categorical methods in proof theory. It is designed to be\u0000accessible, with no prior familiarity with category theory required. The\u0000necessary categorical background is introduced gradually, with a focus on the\u0000philosophical and informal aspects of proof. The only prerequisites are a basic\u0000understanding of logic, computability theory, topology, and ordered structures.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187787","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Nullnorms on bounded trellises 有界花架上的无效模式
arXiv - MATH - Logic Pub Date : 2024-08-18 DOI: arxiv-2408.09321
Zhenyu Xiu, Xu Zheng
{"title":"Nullnorms on bounded trellises","authors":"Zhenyu Xiu, Xu Zheng","doi":"arxiv-2408.09321","DOIUrl":"https://doi.org/arxiv-2408.09321","url":null,"abstract":"In this paper, we introduce the notion of nullnorms on bounded trellises and\u0000study some basic properties. Based on the existence of $t$-norms and\u0000$t$-conorms on arbitrary bounded trellises, we propose some construction\u0000methods of nullnorms on bounded trellises. Moreover, some illustrative examples\u0000are provided.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187790","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A mixed logic with binary operators 二元运算符混合逻辑
arXiv - MATH - Logic Pub Date : 2024-08-18 DOI: arxiv-2408.09581
Ivo Düntsch, Rafał Gruszczyński, Paula Menchón
{"title":"A mixed logic with binary operators","authors":"Ivo Düntsch, Rafał Gruszczyński, Paula Menchón","doi":"arxiv-2408.09581","DOIUrl":"https://doi.org/arxiv-2408.09581","url":null,"abstract":"In previous work \"Betweenness algebras\" we introduced and examined the class\u0000of betweenness algebras. In the current paper we study a larger class of\u0000algebras with binary operators of possibility and sufficiency, the weak mixed\u0000algebras. Furthermore, we develop a system of logic with two binary modalities,\u0000sound and complete with respect to the class of frames closely related to the\u0000aforementioned algebras, and we prove an embedding theorem which solves an open\u0000problem from \"Betweenness algebras\".","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187786","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Measurable Regular Subgraphs 可测量的正则子图
arXiv - MATH - Logic Pub Date : 2024-08-18 DOI: arxiv-2408.09597
Matt Bowen, Clinton T. Conley, Felix Weilacher
{"title":"Measurable Regular Subgraphs","authors":"Matt Bowen, Clinton T. Conley, Felix Weilacher","doi":"arxiv-2408.09597","DOIUrl":"https://doi.org/arxiv-2408.09597","url":null,"abstract":"We show that every $d$-regular bipartite Borel graph admits a Baire\u0000measurable $k$-regular spanning subgraph if and only if $d$ is odd or $k$ is\u0000even. This gives the first example of a locally checkable coloring problem\u0000which is known to have a Baire measurable solution on Borel graphs but not a\u0000computable solution on highly computable graphs. We also prove the analogous\u0000result in the measure setting for hyperfinite graphs.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187788","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Information vs Dimension -- an Algorithmic Perspective 信息与维度--算法视角
arXiv - MATH - Logic Pub Date : 2024-08-09 DOI: arxiv-2408.05121
Jan Reimann
{"title":"Information vs Dimension -- an Algorithmic Perspective","authors":"Jan Reimann","doi":"arxiv-2408.05121","DOIUrl":"https://doi.org/arxiv-2408.05121","url":null,"abstract":"This paper surveys work on the relation between fractal dimensions and\u0000algorithmic information theory over the past thirty years. It covers the basic\u0000development of prefix-free Kolmogorov complexity from an information theoretic\u0000point of view, before introducing Hausdorff measures and dimension along with\u0000some important examples. The main goal of the paper is to motivate and develop\u0000the informal identity \"entropy = complexity = dimension\" from first principles.\u0000The last section of the paper presents some new observations on multifractal\u0000measures from an algorithmic viewpoint.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141945518","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","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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