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Bulletin of Symbolic Logic最新文献

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Kurt Gödel's First Steps in Logic: Formal Proofs in Arithmetic and Set Theory through a System of Natural Deduction 库尔特Gödel逻辑的第一步:通过自然演绎系统的算术和集合论的形式证明
IF 0.6 3区 数学
Bulletin of Symbolic Logic Pub Date : 2018-09-01 DOI: 10.1017/BSL.2017.42
J. Plato
{"title":"Kurt Gödel's First Steps in Logic: Formal Proofs in Arithmetic and Set Theory through a System of Natural Deduction","authors":"J. Plato","doi":"10.1017/BSL.2017.42","DOIUrl":"https://doi.org/10.1017/BSL.2017.42","url":null,"abstract":"What seem to be Kurt Gödel’s first notes on logic, an exercise notebook of 84 pages, contains formal proofs in higher-order arithmetic and set theory. The choice of these topics is clearly suggested by their inclusion in Hilbert and Ackermann’s logic book of 1928, the Grundzüge der theoretischen Logik. Such proofs are notoriously hard to construct within axiomatic logic. Gödel takes without further ado into use a linear system of natural deduction for the full language of higher-order logic, with formal derivations closer to one hundred steps in length and up to four nested temporary assumptions with their scope indicated by vertical intermittent lines.","PeriodicalId":55307,"journal":{"name":"Bulletin of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2018-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80434226","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}
引用次数: 2
The Ramified analytical Hierarchy using Extended Logics 应用扩展逻辑的分支分析层次结构
IF 0.6 3区 数学
Bulletin of Symbolic Logic Pub Date : 2018-08-11 DOI: 10.1017/BSL.2018.69
P. Welch
{"title":"The Ramified analytical Hierarchy using Extended Logics","authors":"P. Welch","doi":"10.1017/BSL.2018.69","DOIUrl":"https://doi.org/10.1017/BSL.2018.69","url":null,"abstract":"The use of Extended Logics to replace ordinary second order definability in Kleene’s Ramified Analytical Hierarchy is investigated. This mirrors a similar investigation of Kennedy, Magidor and Väänänen [11] where Gödel’s universe L of constructible sets is subjected to similar variance. Enhancing second order definability allows models to be defined which may or may not coincide with the original Kleene hierarchy in domain. Extending the logic with game quantifiers, and assuming strong axioms of infinity, we obtain minimal correct models of analysis. A wide spectrum of models can be so generated from abstract definability notions: one may take an abstract Spector Class and extract an extended logic for it. The resultant structure is then a minimal model of the given kind of definability.","PeriodicalId":55307,"journal":{"name":"Bulletin of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2018-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83884846","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}
引用次数: 0
A Hierarchy of computably Enumerable Degrees 可计算可枚举度的层次结构
IF 0.6 3区 数学
Bulletin of Symbolic Logic Pub Date : 2018-03-01 DOI: 10.1017/BSL.2017.41
R. Downey, Noam Greenberg
{"title":"A Hierarchy of computably Enumerable Degrees","authors":"R. Downey, Noam Greenberg","doi":"10.1017/BSL.2017.41","DOIUrl":"https://doi.org/10.1017/BSL.2017.41","url":null,"abstract":"We introduce a new hierarchy of computably enumerable degrees. This hierarchy is based on computable ordinal notations measuring complexity of approximation of ∆2 functions. The hierarchy unifies and classifies the combinatorics of a number of diverse constructions in computability theory. It does so along the lines of the high degrees (Martin) and the array noncomputable degrees (Downey, Jocksuch and Stob). The hierarchy also gives a number of natural definability results in the c.e. degrees, including a definable","PeriodicalId":55307,"journal":{"name":"Bulletin of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2018-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80048800","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}
引用次数: 4
Alexander Razborov, Flag algebras. Journal of Symbolic Logic, vol. 72 (2007), no. 4, pp. 1239-1282 Alexander Razborov，旗代数。符号逻辑学报，vol. 72 (2007)， no。4，第1239-1282页
IF 0.6 3区 数学
Bulletin of Symbolic Logic Pub Date : 2018-01-01 DOI: 10.1017/bsl.2018.11
J. Cummings
{"title":"Alexander Razborov, Flag algebras. Journal of Symbolic Logic, vol. 72 (2007), no. 4, pp. 1239-1282","authors":"J. Cummings","doi":"10.1017/bsl.2018.11","DOIUrl":"https://doi.org/10.1017/bsl.2018.11","url":null,"abstract":"","PeriodicalId":55307,"journal":{"name":"Bulletin of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78102250","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}
引用次数: 0
Polymorphism and the obstinate circularity of second order logic: a victims' tale 多态性和二阶逻辑的顽固循环:一个受害者的故事
IF 0.6 3区 数学
Bulletin of Symbolic Logic Pub Date : 2017-10-27 DOI: 10.1017/bsl.2017.43
Paolo Pistone
{"title":"Polymorphism and the obstinate circularity of second order logic: a victims' tale","authors":"Paolo Pistone","doi":"10.1017/bsl.2017.43","DOIUrl":"https://doi.org/10.1017/bsl.2017.43","url":null,"abstract":"The investigations on higher-order type theories and on the related notion of parametric polymorphism constitute the technical counterpart of the old foundational problem of the circularity (or impredicativity) of second and higher order logic. However, the epistemological significance of such investigations, and of their often non trivial results, has not received much attention in the contemporary foundational debate. The results recalled in this paper suggest that the question of the circularity of second order logic cannot be reduced to the simple assessment of a vicious circle. Through a comparison between the faulty consistency arguments given by Frege and Martin-L\"of, respectively for the logical system of the Grundgesetze (shown inconsistent by Russell's paradox) and for the intuitionistic type theory with a type of all types (shown inconsistent by Girard's paradox), and the normalization argument for second order type theory (or System F), we indicate a bunch of subtle mathematical problems and logical concepts hidden behind the hazardous idea of impredicative quantification, constituting a vast (and largely unexplored) domain for foundational research.","PeriodicalId":55307,"journal":{"name":"Bulletin of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2017-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86060425","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}
引用次数: 5
A Constructive Examination of a Russell-Style Ramified Type Theory 罗素式分支类型理论的建构性检验
IF 0.6 3区 数学
Bulletin of Symbolic Logic Pub Date : 2017-04-22 DOI: 10.1017/bsl.2018.4
Erik Palmgren
{"title":"A Constructive Examination of a Russell-Style Ramified Type Theory","authors":"Erik Palmgren","doi":"10.1017/bsl.2018.4","DOIUrl":"https://doi.org/10.1017/bsl.2018.4","url":null,"abstract":"In this paper we examine the natural interpretation of a ramified type hierarchy into Martin-L\"of type theory with an infinite sequence of universes. It is shown that under this predicative interpretation some useful special cases of Russell's reducibility axiom are valid, namely functional reducibility. This is sufficient to make the type hierarchy usable for development of constructive mathematics. We present a ramified type theory suitable for this purpose. One may regard the results of this paper as an alternative solution to the problems of Russell's theory, which avoids impredicativity, but instead imposes constructive logic. The intuitionistic ramified type theory introduced here, also suggests that there is a natural associated notion of predicative elementary topos.","PeriodicalId":55307,"journal":{"name":"Bulletin of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2017-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86320932","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}
引用次数: 5
In Search of ultimate-l the 19th Midrasha Mathematicae Lectures 追寻终极——第19届米德拉夏数学讲座
IF 0.6 3区 数学
Bulletin of Symbolic Logic Pub Date : 2017-03-01 DOI: 10.1017/BSL.2016.34
W. Woodin
{"title":"In Search of ultimate-l the 19th Midrasha Mathematicae Lectures","authors":"W. Woodin","doi":"10.1017/BSL.2016.34","DOIUrl":"https://doi.org/10.1017/BSL.2016.34","url":null,"abstract":"We give a fairly complete account which first shows that the solution to the inner model problem for one supercompact cardinal will yield an ultimate version of L and then shows that the various current approaches to inner model theory must be fundamentally altered to provide that solution.","PeriodicalId":55307,"journal":{"name":"Bulletin of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2017-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81181982","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}
引用次数: 58
G.O. Jones and A.J. Wilkie, editors, O-Minimality and Diophantine Geometry. London Mathematical Society Lecture Note Series, vol. 421, Cambridge University Press, 2015. xii + 221 pp
IF 0.6 3区 数学
Bulletin of Symbolic Logic Pub Date : 2017-03-01 DOI: 10.1017/BSL.2017.3
Antoine Chambert-Loir
{"title":"G.O. Jones and A.J. Wilkie, editors, O-Minimality and Diophantine Geometry. London Mathematical Society Lecture Note Series, vol. 421, Cambridge University Press, 2015. xii + 221 pp","authors":"Antoine Chambert-Loir","doi":"10.1017/BSL.2017.3","DOIUrl":"https://doi.org/10.1017/BSL.2017.3","url":null,"abstract":"","PeriodicalId":55307,"journal":{"name":"Bulletin of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2017-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75661741","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}
引用次数: 0
In Memoriam: Walter (Wouter) van Stigt (1927-2015)
IF 0.6 3区 数学
Bulletin of Symbolic Logic Pub Date : 2017-03-01 DOI: 10.1017/BSL.2016.39
W. Veldman
{"title":"In Memoriam: Walter (Wouter) van Stigt (1927-2015)","authors":"W. Veldman","doi":"10.1017/BSL.2016.39","DOIUrl":"https://doi.org/10.1017/BSL.2016.39","url":null,"abstract":"","PeriodicalId":55307,"journal":{"name":"Bulletin of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2017-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/BSL.2016.39","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72374617","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}
引用次数: 0
Robert I. Soare, Turing Computability, Theory and Applications of Computability, Springer-Verlag, Berlin, Heidelberg, 2016, xxxvi + 263 pp Robert I. Soare，图灵可计算性，理论与应用，Springer-Verlag，柏林，海德堡，2016,xxxvi + 263页
IF 0.6 3区 数学
Bulletin of Symbolic Logic Pub Date : 2017-03-01 DOI: 10.1017/bsl.2017.1
D. Dzhafarov
{"title":"Robert I. Soare, Turing Computability, Theory and Applications of Computability, Springer-Verlag, Berlin, Heidelberg, 2016, xxxvi + 263 pp","authors":"D. Dzhafarov","doi":"10.1017/bsl.2017.1","DOIUrl":"https://doi.org/10.1017/bsl.2017.1","url":null,"abstract":"","PeriodicalId":55307,"journal":{"name":"Bulletin of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2017-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74705157","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}
引用次数: 0
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