Pub Date : 2021-01-01DOI: 10.1215/00294527-2021-0001
Emmylou Haffner
In this paper, I propose to examine Dedekind’s ideal of rigor in the context of some of his mathematical drafts. After a presentation of his ideal of rigor based on statements in his published works, I use drafts from his Nachlass to study his invention of the new concept of Dualgruppe (equivalent to our modern lattice). I question the extent to which these preliminary researches hold up to the same standards of rigor. Focusing on a specific law of Dualgruppe theory, I show that the elaboration of a rigorous work can be the outcome of a process that is not necessarily so. I put forward the trial-and-error and inductive aspects of Dedekind’s research practices. I consider whether the Dedekindian ideal of rigor guided mathematical research in its various phases, and what the consequences were of such an ideal of rigor, if any, on mathematical research.
{"title":"The Shaping of Dedekind's Rigorous Mathematics: What Do Dedekind's Drafts Tell Us about His Ideal of Rigor?","authors":"Emmylou Haffner","doi":"10.1215/00294527-2021-0001","DOIUrl":"https://doi.org/10.1215/00294527-2021-0001","url":null,"abstract":"In this paper, I propose to examine Dedekind’s ideal of rigor in the context of some of his mathematical drafts. After a presentation of his ideal of rigor based on statements in his published works, I use drafts from his Nachlass to study his invention of the new concept of Dualgruppe (equivalent to our modern lattice). I question the extent to which these preliminary researches hold up to the same standards of rigor. Focusing on a specific law of Dualgruppe theory, I show that the elaboration of a rigorous work can be the outcome of a process that is not necessarily so. I put forward the trial-and-error and inductive aspects of Dedekind’s research practices. I consider whether the Dedekindian ideal of rigor guided mathematical research in its various phases, and what the consequences were of such an ideal of rigor, if any, on mathematical research.","PeriodicalId":51259,"journal":{"name":"Notre Dame Journal of Formal Logic","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91090743","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-01DOI: 10.1215/00294527-2021-0005
Paul Anh Tran-Hoang
{"title":"On the Virtue of Categoricity","authors":"Paul Anh Tran-Hoang","doi":"10.1215/00294527-2021-0005","DOIUrl":"https://doi.org/10.1215/00294527-2021-0005","url":null,"abstract":"","PeriodicalId":51259,"journal":{"name":"Notre Dame Journal of Formal Logic","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80318046","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-01DOI: 10.1215/00294527-2021-0007
Stewart Shapiro, C. Roberts
{"title":"Open Texture and Mathematics","authors":"Stewart Shapiro, C. Roberts","doi":"10.1215/00294527-2021-0007","DOIUrl":"https://doi.org/10.1215/00294527-2021-0007","url":null,"abstract":"","PeriodicalId":51259,"journal":{"name":"Notre Dame Journal of Formal Logic","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82748098","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-12-01DOI: 10.1215/00294527-2020-0022
T. Ahmed
Fix 2 < n < ω. Let CAn denotes the class of cylindric algebras of dimension n, and RCAn denotes the variety of representable CAns. Let Ln denote rst order logic restricted to the rst n variables. Roughly CAn, an instance of Boolean algbras with operators, is the algebraic counterpart of the syntax of Ln, namely, its proof theory, while RCAn represents algebraically and geometrically Tarskian semantics of Ln. Unlike Boolean algebras, having a Stone representation theorem, RCAn ( CAn. Using combinatorial game theory, we show that the existence of certain nite relation algebras RAs, which are algebras whose domain consists of binary relations, imply that the celebrated Henkin omitting types theorem, fails in a very strong sense for Ln. Using special cases of such nite RAs, we recover the classical nonnite axiomatizability results of Monk, Maddux and Biro on RCAn and we reprove Hirsch and Hodkinson's result that the class of completely representable CAns is not rst order de nable. We show that if T is an Ln countable theory that admits elimination of quanti ers, λ is a cardinal < 2א0 and F = ⟨Γi : i < λ⟩ is a family of complete non-principal types, then F can be omitted in an ordinary countable model of T .
固定2 < n < ω。设CAn表示维数为n的圆柱代数的类，RCAn表示可表示的CAn的种类。设Ln表示一阶逻辑，限制于第n个变量。粗略地说，CAn是带算子的布尔代数的一个实例，它是Ln语法的代数对应，即它的证明理论，而RCAn则表示Ln的代数和几何塔斯基语义。不像布尔代数，有一个斯通表示定理，RCAn (CAn。利用组合博弈论，我们证明了某些定义域由二元关系组成的整数关系代数RAs的存在性，这意味着著名的Henkin省略型定理在Ln上是不成立的。利用这类非整数可表示性的特例，我们恢复了Monk、Maddux和Biro关于RCAn的经典非整数公化性结果，并证明了Hirsch和Hodkinson关于完全可表示性的can类不是一阶可表示性的结论。我们证明，如果T是一个允许消除量数的Ln可数理论，λ是基数< 2 μ l，并且F =⟨Γi: i < λ⟩是一个完全的非主类型族，那么F可以在T的普通可数模型中省略。
{"title":"Existence of Certain Finite Relation Algebras Implies Failure of Omitting Types for L n","authors":"T. Ahmed","doi":"10.1215/00294527-2020-0022","DOIUrl":"https://doi.org/10.1215/00294527-2020-0022","url":null,"abstract":"Fix 2 < n < ω. Let CAn denotes the class of cylindric algebras of dimension n, and RCAn denotes the variety of representable CAns. Let Ln denote rst order logic restricted to the rst n variables. Roughly CAn, an instance of Boolean algbras with operators, is the algebraic counterpart of the syntax of Ln, namely, its proof theory, while RCAn represents algebraically and geometrically Tarskian semantics of Ln. Unlike Boolean algebras, having a Stone representation theorem, RCAn ( CAn. Using combinatorial game theory, we show that the existence of certain nite relation algebras RAs, which are algebras whose domain consists of binary relations, imply that the celebrated Henkin omitting types theorem, fails in a very strong sense for Ln. Using special cases of such nite RAs, we recover the classical nonnite axiomatizability results of Monk, Maddux and Biro on RCAn and we reprove Hirsch and Hodkinson's result that the class of completely representable CAns is not rst order de nable. We show that if T is an Ln countable theory that admits elimination of quanti ers, λ is a cardinal < 2א0 and F = ⟨Γi : i < λ⟩ is a family of complete non-principal types, then F can be omitted in an ordinary countable model of T .","PeriodicalId":51259,"journal":{"name":"Notre Dame Journal of Formal Logic","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49044957","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-11-29DOI: 10.1215/00294527-2021-0029
Philip Dittmann, Arno Fehm
We show that the set of algebraic extensions $F$ of $mathbb{Q}$ in which $mathbb{Z}$ or the ring of integers $mathcal{O}_F$ are definable is meager in the set of all algebraic extensions.
{"title":"Nondefinability of Rings of Integers in Most Algebraic Fields","authors":"Philip Dittmann, Arno Fehm","doi":"10.1215/00294527-2021-0029","DOIUrl":"https://doi.org/10.1215/00294527-2021-0029","url":null,"abstract":"We show that the set of algebraic extensions $F$ of $mathbb{Q}$ in which $mathbb{Z}$ or the ring of integers $mathcal{O}_F$ are definable is meager in the set of all algebraic extensions.","PeriodicalId":51259,"journal":{"name":"Notre Dame Journal of Formal Logic","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2020-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79098815","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-11-01DOI: 10.1215/00294527-2020-0028
Ryo Kashima, Naosuke Matsuda, Takao Yuyama
Barendregt gave a sound semantics of the simple type assignment system λ→ by generalizing Tait’s proof of the strong normalization theorem. In this paper, we aim to extend the semantics so that the completeness theorem holds.
{"title":"Term-Space Semantics of Typed Lambda Calculus","authors":"Ryo Kashima, Naosuke Matsuda, Takao Yuyama","doi":"10.1215/00294527-2020-0028","DOIUrl":"https://doi.org/10.1215/00294527-2020-0028","url":null,"abstract":"Barendregt gave a sound semantics of the simple type assignment system λ→ by generalizing Tait’s proof of the strong normalization theorem. In this paper, we aim to extend the semantics so that the completeness theorem holds.","PeriodicalId":51259,"journal":{"name":"Notre Dame Journal of Formal Logic","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2020-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48021065","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-11-01DOI: 10.1215/00294527-2020-0023
Yong Liu, Cheng Peng
We show that for any ω-r.e. degree d and n-r.e. degree b with d
我们证明了对于任意ω- re。d阶和n阶。当d
{"title":"Weak Density and Nondensity among Transfinite Levels of the Ershov Hierarchy","authors":"Yong Liu, Cheng Peng","doi":"10.1215/00294527-2020-0023","DOIUrl":"https://doi.org/10.1215/00294527-2020-0023","url":null,"abstract":"We show that for any ω-r.e. degree d and n-r.e. degree b with d<b, there is an (ω+1)-r.e. degree a strictly between d and b. We also show that there is a maximal incomplete (ω+1)-r.e. degree. As a corollary, Dω is not a Σ1-elementary substructure of Dω+1.","PeriodicalId":51259,"journal":{"name":"Notre Dame Journal of Formal Logic","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2020-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48213150","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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