We discuss several ontological properties of explicit mathematics and operational set theory: global choice, decidable classes, totality and extensionality of operations, function spaces, class and set formation via formulas that contain the definedness predicate and applications.
{"title":"Explicit Mathematics and Operational Set Theory: some Ontological comparisons","authors":"Gerhard Jäger, Rico Zumbrunnen","doi":"10.1017/BSL.2014.21","DOIUrl":"https://doi.org/10.1017/BSL.2014.21","url":null,"abstract":"We discuss several ontological properties of explicit mathematics and operational set theory: global choice, decidable classes, totality and extensionality of operations, function spaces, class and set formation via formulas that contain the definedness predicate and applications.","PeriodicalId":55307,"journal":{"name":"Bulletin of Symbolic Logic","volume":"13 1","pages":"275-292"},"PeriodicalIF":0.6,"publicationDate":"2014-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75355604","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}
{"title":"David Hilbert's lectures on the foundations of arithmetic and logic 1917-1933. Edited by William Ewald and Wilfried Sieg. Springer, Berlin, Heidelberg and New York, 2013, xxv + 1062 pp","authors":"J. Plato","doi":"10.1017/BSL.2014.27","DOIUrl":"https://doi.org/10.1017/BSL.2014.27","url":null,"abstract":"","PeriodicalId":55307,"journal":{"name":"Bulletin of Symbolic Logic","volume":"130 1","pages":"363-365"},"PeriodicalIF":0.6,"publicationDate":"2014-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73807591","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}
We provide an introduction to methods and recent results on infinitely generated abelian groups with decidable word problem.
介绍了具有可决字问题的无限生成阿贝尔群的研究方法和最新结果。
{"title":"Computable Abelian Groups","authors":"A. Melnikov","doi":"10.1017/BSL.2014.32","DOIUrl":"https://doi.org/10.1017/BSL.2014.32","url":null,"abstract":"We provide an introduction to methods and recent results on infinitely generated abelian groups with decidable word problem.","PeriodicalId":55307,"journal":{"name":"Bulletin of Symbolic Logic","volume":"1 1","pages":"315-356"},"PeriodicalIF":0.6,"publicationDate":"2014-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83641804","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}
{"title":"Recent developments in model theory, Notre Dame Journal of Formal Logic, vol.54, nos.3-4, 2013","authors":"D. Macpherson","doi":"10.1017/bsl.2014.25","DOIUrl":"https://doi.org/10.1017/bsl.2014.25","url":null,"abstract":"","PeriodicalId":55307,"journal":{"name":"Bulletin of Symbolic Logic","volume":"14 1","pages":"357-359"},"PeriodicalIF":0.6,"publicationDate":"2014-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90059562","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}
The history of productivity of the κ -chain condition in partial orders, topological spaces, or Boolean algebras is surveyed, and its connection to the set-theoretic notion of a weakly compact cardinal is highlighted. Then, it is proved that for every regular cardinal $kappa > aleph _1 {rm{,}}$ the principle □( k ) is equivalent to the existence of a certain strong coloring $c,:,[k]^2 , to $ k for which the family of fibers ${cal T}left( c right)$ is a nonspecial κ -Aronszajn tree. The theorem follows from an analysis of a new characteristic function for walks on ordinals, and implies in particular that if the κ -chain condition is productive for a given regular cardinal $kappa > aleph _1 {rm{,}}$ then κ is weakly compact in some inner model of ZFC. This provides a partial converse to the fact that if κ is a weakly compact cardinal, then the κ -chain condition is productive.
研究了k -链条件在偏序、拓扑空间或布尔代数中的生产力的历史，并强调了它与弱紧基数的集合论概念的联系。然后，证明了对于每一个正则基$kappa > aleph _1 {rm{,}}$，原理□(k)等价于一个强着色$c,:,[k]^2 , to $ k的存在性，对于该强着色 k，纤维族${cal T}left( c right)$是一个非特殊的κ -Aronszajn树。该定理来源于对序上行走的一个新的特征函数的分析，并特别表明，如果κ链条件对于给定的正则基数$kappa > aleph _1 {rm{,}}$是有效的，则κ在ZFC的某个内模型中是弱紧的。这提供了一个部分相反的事实，如果κ是一个弱紧基数，那么κ链条件是有效的。
{"title":"Chain conditions of Products, and Weakly Compact Cardinals","authors":"A. Rinot","doi":"10.1017/BSL.2014.24","DOIUrl":"https://doi.org/10.1017/BSL.2014.24","url":null,"abstract":"The history of productivity of the κ -chain condition in partial orders, topological spaces, or Boolean algebras is surveyed, and its connection to the set-theoretic notion of a weakly compact cardinal is highlighted. Then, it is proved that for every regular cardinal $kappa > aleph _1 {rm{,}}$\u0000 the principle □( k ) is equivalent to the existence of a certain strong coloring $c,:,[k]^2 , to $\u0000 k for which the family of fibers ${cal T}left( c right)$\u0000 is a nonspecial κ -Aronszajn tree. The theorem follows from an analysis of a new characteristic function for walks on ordinals, and implies in particular that if the κ -chain condition is productive for a given regular cardinal $kappa > aleph _1 {rm{,}}$\u0000 then κ is weakly compact in some inner model of ZFC. This provides a partial converse to the fact that if κ is a weakly compact cardinal, then the κ -chain condition is productive.","PeriodicalId":55307,"journal":{"name":"Bulletin of Symbolic Logic","volume":"32 1","pages":"293-314"},"PeriodicalIF":0.6,"publicationDate":"2014-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80389128","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}
{"title":"Ulrich Pardey, Frege on Absolute and Relative Truth. An Introduction to the Practice of Interpreting Philosophical Texts, History of Analytic Philosophy (no vol. number), Palgrave Macmillian, Basingstoke-New York, 2012, xxiv+242 pp","authors":"B. Buldt","doi":"10.1017/BSL.2014.26","DOIUrl":"https://doi.org/10.1017/BSL.2014.26","url":null,"abstract":"","PeriodicalId":55307,"journal":{"name":"Bulletin of Symbolic Logic","volume":"75 1","pages":"360-362"},"PeriodicalIF":0.6,"publicationDate":"2014-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85759603","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}
Euclidean geometry consists of straightedge-and-compass constructions and reasoning about the results of those constructions. We show that Euclidean geometry can be developed using only intuitionistic logic. We consider three versions of Euclid's parallel postulate: Euclid's own formulation in his Postulate 5; Playfair's 1795 version, and a new version we call the strong parallel postulate. These differ in that Euclid's version and the new version both assert the existence of a point where two lines meet, while Playfair's version makes no existence assertion. Classically, the models of Euclidean (straightedge-and-compass) geometry are planes over Euclidean fields. We prove a similar theorem for constructive Euclidean geometry, by showing how to define addition and multiplication without a case distinction about the sign of the arguments. With intuitionistic logic, there are two possible definitions of Euclidean fields, which turn out to correspond to the different versions of the parallel axiom. In this paper, we completely settle the questions about implications between the three versions of the parallel postulate: the strong parallel postulate easily implies Euclid 5, and in fact Euclid 5 also implies the strong parallel postulate, although the proof is lengthy, depending on the verification that Euclid 5 suffices to define multiplication geometrically. We show that Playfair does not imply Euclid 5, and we also give some other independence results. Our independence proofs are given without discussing the exact choice of the other axioms of geometry; all we need is that one can interpret the geometric axioms in Euclidean field theory. The proofs use Kripke models of Euclidean field theories based on carefully constructed rings of real-valued functions.
{"title":"Constructive Geometry and the Parallel postulate","authors":"M. Beeson","doi":"10.1017/BSL.2015.41","DOIUrl":"https://doi.org/10.1017/BSL.2015.41","url":null,"abstract":"Euclidean geometry consists of straightedge-and-compass constructions and reasoning about the results of those constructions. We show that Euclidean geometry can be developed using only intuitionistic logic. We consider three versions of Euclid's parallel postulate: Euclid's own formulation in his Postulate 5; Playfair's 1795 version, and a new version we call the strong parallel postulate. These differ in that Euclid's version and the new version both assert the existence of a point where two lines meet, while Playfair's version makes no existence assertion. Classically, the models of Euclidean (straightedge-and-compass) geometry are planes over Euclidean fields. We prove a similar theorem for constructive Euclidean geometry, by showing how to define addition and multiplication without a case distinction about the sign of the arguments. With intuitionistic logic, there are two possible definitions of Euclidean fields, which turn out to correspond to the different versions of the parallel axiom. In this paper, we completely settle the questions about implications between the three versions of the parallel postulate: the strong parallel postulate easily implies Euclid 5, and in fact Euclid 5 also implies the strong parallel postulate, although the proof is lengthy, depending on the verification that Euclid 5 suffices to define multiplication geometrically. We show that Playfair does not imply Euclid 5, and we also give some other independence results. Our independence proofs are given without discussing the exact choice of the other axioms of geometry; all we need is that one can interpret the geometric axioms in Euclidean field theory. The proofs use Kripke models of Euclidean field theories based on carefully constructed rings of real-valued functions.","PeriodicalId":55307,"journal":{"name":"Bulletin of Symbolic Logic","volume":"1 1","pages":"1-104"},"PeriodicalIF":0.6,"publicationDate":"2014-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90582482","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}
{"title":"Nicholas J. J. Smith. Logic: The laws of truth. Princeton University Press, Princeton NJ, 2012, xiv + 528 pp","authors":"Jennifer Duke-Yonge","doi":"10.1017/BSL.2014.19","DOIUrl":"https://doi.org/10.1017/BSL.2014.19","url":null,"abstract":"","PeriodicalId":55307,"journal":{"name":"Bulletin of Symbolic Logic","volume":"1 1","pages":"202-203"},"PeriodicalIF":0.6,"publicationDate":"2014-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74645987","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}
We give a survey of the study of nonstandard models in recursion theory and reverse mathematics. We discuss the key notions and techniques in effective computability in nonstandard models, and their applications to problems concerning combinatorial principles in subsystems of second order arithmetic. Particular attention is given to principles related to Ramsey’s Theorem for Pairs.
{"title":"Nonstandard Models in Recursion Theory and Reverse Mathematics","authors":"C. Chong, Wei Li, Yue Yang","doi":"10.1017/BSL.2014.14","DOIUrl":"https://doi.org/10.1017/BSL.2014.14","url":null,"abstract":"We give a survey of the study of nonstandard models in recursion theory and reverse mathematics. We discuss the key notions and techniques in effective computability in nonstandard models, and their applications to problems concerning combinatorial principles in subsystems of second order arithmetic. Particular attention is given to principles related to Ramsey’s Theorem for Pairs.","PeriodicalId":55307,"journal":{"name":"Bulletin of Symbolic Logic","volume":"95 1","pages":"170-200"},"PeriodicalIF":0.6,"publicationDate":"2014-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85376188","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}
{"title":"Approximating Beppo Levi's principio di Approssimazione","authors":"Riccardo Bruni, Peter Schuster","doi":"10.1017/BSL.2014.6","DOIUrl":"https://doi.org/10.1017/BSL.2014.6","url":null,"abstract":"","PeriodicalId":55307,"journal":{"name":"Bulletin of Symbolic Logic","volume":"279 1","pages":"141-169"},"PeriodicalIF":0.6,"publicationDate":"2014-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72750894","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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