The Bulletin of Symbolic Logic最新文献_第8页

The Bulletin of Symbolic Logic Pub Date : 2021-12-01 DOI: 10.1017/bsl.2022.4

Lajos Soukup, Grigor Sargsyan, Laurent Bienvenu, Barbara F. Csima, Daniel T. Soukup, Taishi Kurahashi, Yoshihiro Maruyama, Osvaldo Guzmán, J. Joosten, S. Terwijn, Takayuki Kihara

引用次数: 0

BSL volume 27 issue 4 Cover and Front matter BSL第27卷第4期封面和封面问题

The Bulletin of Symbolic Logic Pub Date : 2021-12-01 DOI: 10.1017/bsl.2022.5

引用次数: 0

Some Progress on the Unique Ergodicity Problem 关于唯一遍历性问题的一些进展

The Bulletin of Symbolic Logic Pub Date : 2021-12-01 DOI: 10.1017/bsl.2021.63

Colin Jahel

{"title":"Some Progress on the Unique Ergodicity Problem","authors":"Colin Jahel","doi":"10.1017/bsl.2021.63","DOIUrl":"https://doi.org/10.1017/bsl.2021.63","url":null,"abstract":"Abstract This thesis is at the intersection of dynamics, probability and model theory. It focuses on a specialization of the notion of amenability: unique ergodicity. Let G be a Polish group, i.e., a topological group whose topology is separable and completely metrizable. We call a G-flow the action of G on a compact space. A G-flow is said to be minimal if every orbit is dense. A famous theorem of Ellis states that any Polish group G admits a unique universal minimal flow that we denote \u0000${mathrm {M}}(G)$\u0000 . This means that for any minimal G-flow X there is a surjective G-map from \u0000${mathrm {M}}(G)$\u0000 to X. G is said to be amenable if every G-flow admits an invariant probability measure, and uniquely ergodic if every minimal flow admits a unique invariant probability measure. The notion of unique ergodicity relating to a group was introduced by Angel, Kechris and Lyons. They also ask the following question which is the main focus of the thesis: Let G be an amenable Polish group with metrizable universal minimal flow, is G uniquely ergodic? Note that unique ergodicity is an interesting notion only for relatively large groups, as it is proved in the last chapter of this thesis that locally compact non compact Polish groups are never uniquely ergodic. This result is joint work with Andy Zucker. The thesis includes proofs of unique ergodicity of groups with interesting universal minimal flows, namely the automorphism group of the semigeneric directed graph and the automorphism group of the \u0000$2$\u0000 -graph. It also includes a theorem stating that under some hypothesis on a \u0000$omega $\u0000 -categorical structure M, the logic action of \u0000${mathrm {Aut}}(M)$\u0000 on \u0000${mathrm {LO}}(M)$\u0000 , the compact space of linear orders on M, is uniquely ergodic. This implies unique ergodicity for the group if its universal minimal flow happens to be the space of linear orderings. It can also be used to prove non-amenability of some groups for which the action of \u0000${mathrm {Aut}}(M)$\u0000 on \u0000${mathrm {LO}}(M)$\u0000 is not minimal. This result is joint work with Todor Tsankov. Finally, the thesis also presents a proof that under the assumption that the universal minimal flows involved are metrizable, unique ergodicity is stable under group extensions. This result is joint work with Andy Zucker. Abstract prepared by Colin Jahel. E-mail: cjahel@andrew.cmu.edu URL: http://math.univ-lyon1.fr/~jahel/doc/These.pdf","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":"1 1","pages":"527 - 528"},"PeriodicalIF":0.0,"publicationDate":"2021-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76772958","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: 1

IN MEMORIAM: MICHAEL MORLEY, 1930–2020 纪念:迈克尔·莫利，1930-2020

The Bulletin of Symbolic Logic Pub Date : 2021-12-01 DOI: 10.1017/bsl.2021.57

John Baldwin, D. Marker

{"title":"IN MEMORIAM: MICHAEL MORLEY, 1930–2020","authors":"John Baldwin, D. Marker","doi":"10.1017/bsl.2021.57","DOIUrl":"https://doi.org/10.1017/bsl.2021.57","url":null,"abstract":"Michael Darwin Morley, aged 90, Emeritus Professor of Mathematics at Cornell University, passed away October 12, 2020. Morley’s groundbreaking 1965 paper Categoricity in Power was the beginning of modern model theory and laid the foundation for decades of future developments. Morley was born September 29, 1930 in Youngstown, Ohio and received his B.S. degree in mathematics from Case Institute of Technology in 1951. In 1952, he began graduate work at the University of Chicago joining an energetic group of young logicians including William Howard, John Myhill, Anil Nerode, Raymond Smullyan, Stanley Tennenbaum, and the undergraduate Paul Cohen. While at the University of Chicago he met his future wife Vivienne Brenner, a fellow graduate student who finished her thesis on singular integrals under Antoni Zygmund in 1956. They were a devoted couple for over 50 years. Saunders Mac Lane served as his formal advisor at the University of Chicago. Mac Lane refused to grant a Ph.D. for Morley’s early work on saturated models, but helped arrange for Morley’s employment from 1955 to 1961 at the University of Chicago’s Laboratory for Applied Sciences, where he considered military applications of mathematics. Much of Morley’s work on saturated models was discovered independently by Robert Vaught and Morley left Chicago in 1961 to work with Vaught at Berkeley, first as a visiting graduate student and later as an Instructor. Together they published their independent development of saturated models in [26]. This paper built on Barni Jónsson’s development of the notion of uncountable universal-homogeneous models in universal algebra. Its innovations included a) replacing substructure with elementary substructure, and thus universally axiomatizable theories with first order theories, b) introducing special models so as to avoid the reliance on using the GCH (introduced by Hausdorff) to study universal models, and c) the general proof of the uniqueness of saturated models in a given regular cardinality.","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":"99 1","pages":"514 - 518"},"PeriodicalIF":0.0,"publicationDate":"2021-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80579660","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

Towards Finding a Lattice that Characterizes the ${>} omega ^2$ -Fickle Recursively Enumerable Turing Degrees 关于寻找表征${>} ω ^2$ -易变递归可枚举图灵度的格

The Bulletin of Symbolic Logic Pub Date : 2021-12-01 DOI: 10.1017/bsl.2021.56

Liling Ko

{"title":"Towards Finding a Lattice that Characterizes the \u0000${>} omega ^2$\u0000 -Fickle Recursively Enumerable Turing Degrees","authors":"Liling Ko","doi":"10.1017/bsl.2021.56","DOIUrl":"https://doi.org/10.1017/bsl.2021.56","url":null,"abstract":"Abstract Given a finite lattice L that can be embedded in the recursively enumerable (r.e.) Turing degrees \u0000$langle mathcal {R}_{mathrm {T}},leq _{mathrm {T}}rangle $\u0000 , we do not in general know how to characterize the degrees \u0000$mathbf {d}in mathcal {R}_{mathrm {T}}$\u0000 below which L can be bounded. The important characterizations known are of the \u0000$L_7$\u0000 and \u0000$M_3$\u0000 lattices, where the lattices are bounded below \u0000$mathbf {d}$\u0000 if and only if \u0000$mathbf {d}$\u0000 contains sets of “fickleness” \u0000$>omega $\u0000 and \u0000$geq omega ^omega $\u0000 respectively. We work towards finding a lattice that characterizes the levels above \u0000$omega ^2$\u0000 , the first non-trivial level after \u0000$omega $\u0000 . We introduced a lattice-theoretic property called “ \u0000$3$\u0000 -directness” to describe lattices that are no “wider” or “taller” than \u0000$L_7$\u0000 and \u0000$M_3$\u0000 . We exhaust the 3-direct lattices L, but they turn out to also characterize the \u0000$>omega $\u0000 or \u0000$geq omega ^omega $\u0000 levels, if L is not already embeddable below all non-zero r.e. degrees. We also considered upper semilattices (USLs) by removing the bottom meet(s) of some 3-direct lattices, but the removals did not change the levels characterized. This leads us to conjecture that a USL characterizes the same r.e. degrees as the lattice on which the USL is based. We discovered three 3-direct lattices besides \u0000$M_3$\u0000 that also characterize the \u0000$geq omega ^omega $\u0000 -levels. Our search for a \u0000$>omega ^2$\u0000 -candidate therefore involves the lattice-theoretic problem of finding lattices that do not contain any of the four \u0000$geq omega ^omega $\u0000 -lattices as sublattices. Abstract prepared by Liling Ko. E-mail: ko.390@osu.edu URL: http://sites.nd.edu/liling-ko/","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":"327 1","pages":"528 - 528"},"PeriodicalIF":0.0,"publicationDate":"2021-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75932645","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

Algebraic and Model Theoretic Properties of O-minimal Exponential Fields o -极小指数场的代数和模型论性质

The Bulletin of Symbolic Logic Pub Date : 2021-12-01 DOI: 10.1017/bsl.2021.64

L. S. Krapp

{"title":"Algebraic and Model Theoretic Properties of O-minimal Exponential Fields","authors":"L. S. Krapp","doi":"10.1017/bsl.2021.64","DOIUrl":"https://doi.org/10.1017/bsl.2021.64","url":null,"abstract":"Abstract An exponential \u0000$exp $\u0000 on an ordered field \u0000$(K,+,-,cdot ,0,1,<)$\u0000 is an order-preserving isomorphism from the ordered additive group \u0000$(K,+,0,<)$\u0000 to the ordered multiplicative group of positive elements \u0000$(K^{>0},cdot ,1,<)$\u0000 . The structure \u0000$(K,+,-,cdot ,0,1,<,exp )$\u0000 is then called an ordered exponential field (cf. [6]). A linearly ordered structure \u0000$(M,<,ldots )$\u0000 is called o-minimal if every parametrically definable subset of M is a finite union of points and open intervals of M. The main subject of this thesis is the algebraic and model theoretic examination of o-minimal exponential fields \u0000$(K,+,-,cdot ,0,1,<,exp )$\u0000 whose exponential satisfies the differential equation \u0000$exp ' = exp $\u0000 with initial condition \u0000$exp (0) = 1$\u0000 . This study is mainly motivated by the Transfer Conjecture, which states as follows: Any o-minimal exponential field \u0000$(K,+,-,cdot ,0,1,<,exp )$\u0000 whose exponential satisfies the differential equation \u0000$exp ' = exp $\u0000 with initial condition \u0000$exp (0)=1$\u0000 is elementarily equivalent to \u0000$mathbb {R}_{exp }$\u0000 . Here, \u0000$mathbb {R}_{exp }$\u0000 denotes the real exponential field \u0000$(mathbb {R},+,-,cdot ,0,1,<,exp )$\u0000 , where \u0000$exp $\u0000 denotes the standard exponential \u0000$x mapsto mathrm {e}^x$\u0000 on \u0000$mathbb {R}$\u0000 . Moreover, elementary equivalence means that any first-order sentence in the language \u0000$mathcal {L}_{exp } = {+,-,cdot ,0,1, <,exp }$\u0000 holds for \u0000$(K,+,-,cdot ,0,1,<,exp )$\u0000 if and only if it holds for \u0000$mathbb {R}_{exp }$\u0000 . The Transfer Conjecture, and thus the study of o-minimal exponential fields, is of particular interest in the light of the decidability of \u0000$mathbb {R}_{exp }$\u0000 . To the date, it is not known if \u0000$mathbb {R}_{exp }$\u0000 is decidable, i.e., whether there exists a procedure determining for a given first-order \u0000$mathcal {L}_{exp }$\u0000 -sentence whether it is true or false in \u0000$mathbb {R}_{exp }$\u0000 . However, under the assumption of Schanuel’s Conjecture—a famous open conjecture from Transcendental Number Theory—a decision procedure for \u0000$mathbb {R}_{exp }$\u0000 exists (cf. [7]). Also a positive answer to the Transfer Conjecture would result in the decidability of \u0000$mathbb {R}_{exp }$\u0000 (cf. [1]). Thus, we study o-minimal exponential fields with regard to the Transfer Conjecture, Schanuel’s Conjecture, and the decidability question of \u0000$mathbb {R}_{exp }$\u0000 . Overall, we shed light on the valuation theoretic invariants of o-minimal exponential fields—the residue field and the value group—with additional induced structure. Moreover, we explore elementary substructures and extensions of o-minimal exponential fields to the maximal ends—the smallest elementary substructures being prime models and the maximal elementary extensions being contained in the surreal numbers. Further, we draw connections to models of Peano Arithmetic, integer parts, density in real closure, definable Henselian valuations, and strongly NIP ordered fields. Parts of this thesis were published in [2–5]. Abstract prepared by Lothar Sebas","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":"2 1","pages":"529 - 530"},"PeriodicalIF":0.0,"publicationDate":"2021-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75136596","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: 8

IN MEMORIAM: J. MICHAEL DUNN, 1941–2021 纪念:j .迈克尔·邓恩，1941-2021

The Bulletin of Symbolic Logic Pub Date : 2021-12-01 DOI: 10.1017/bsl.2021.65

K. Bimbó

{"title":"IN MEMORIAM: J. MICHAEL DUNN, 1941–2021","authors":"K. Bimbó","doi":"10.1017/bsl.2021.65","DOIUrl":"https://doi.org/10.1017/bsl.2021.65","url":null,"abstract":"The history of relevance logic cannot be written without mentioning J. Michael Dunn who played a prominent role in shaping this area of logic. In the late twentieth century, he was a doyen with a world-class reputation in the field of philosophical logic. Dunn’s research also encompassed logics outside of relevance logic, from 2-valued first-order logic to quantum logic and substructural logics such as the Lambek calculi, intuitionistic logic, linear logic, etc. Each of the disciplines of philosophy, mathematics, and computer science has been impacted in intrinsic ways by some of the theorems proved by Dunn. Jon Michael Dunn was born in Fort Wayne, Indiana, on June 19, 1941, and he passed away on April 5, 2021, in Bloomington, Indiana. After attending high schools in Fort Wayne and Lafayette, he obtained an AB degree in philosophy from Oberlin College, Ohio, in 1963. Dunn completed his Ph.D. Thesis entitled The Algebra of Intensional Logics at Pittsburgh University in 1966; his thesis supervisor was Nuel D. Belnap. In 1969, Dunn was appointed an associate professor in the Department of Philosophy at Indiana University in Bloomington, Indiana, and he stayed on the faculty at IU until 2007, when he retired as University Dean of the School of Informatics, Oscar R. Ewing Professor of Philosophy, Professor of Informatics, Professor of Computer Science, and Core Faculty in Cognitive Science. Dunn supervised 14 Ph.D. students in logic and 3 Ph.D. students in other areas of philosophy. He taught advanced graduate courses in logic, including courses on 2-valued logic (metalogic), modal logic, algebraic logics, and substructural logics. Dunn was the founding dean of the School of Informatics, and he served in other administrative positions such as Chair of the Department of Philosophy and Associate Dean of the College of Arts and Sciences in previous years. Dunn held multiple research grants and visiting positions at universities in the US, Europe, and Australia. Dunn, for his contributions to Indiana University, was honored by the IU Provost Medallion in 2007. The state of Indiana bestowed on Dunn the rank and title of Sagamore of the Wabash the same year. Dunn was elected a Fellow of the American Academy of Arts and Sciences in 2010. The logic of relevant implication R combines Church’s “weak implication” with lattice connectives and De Morgan negation. Alternatively, R results from Ackermann’s system Π′ by omitting a rule (the so-called rule) and adding permutation. Dunn started to investigate R from an algebraic point of view in his Ph.D. thesis [14]. This research continued the study of distributive lattices with De Morgan negation already underway in [1, 4, 5, 35, 43]. Dunn showed that 4, the four-element lattice with two incomparable elements on which negation has fixed points, plays a fundamental role among De Morgan lattices, and hence, for first-degree entailments fde; (the implication-free fragment of R and of the logic of entailment E). Dunn pr","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":"54 1","pages":"519 - 525"},"PeriodicalIF":0.0,"publicationDate":"2021-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91252895","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 SENSE/REFERENCE DISTINCTION IN CONSTRUCTIVE SEMANTICS 建构语义学中意义/指称的区别

The Bulletin of Symbolic Logic Pub Date : 2021-12-01 DOI: 10.1017/bsl.2021.61

P. Martin-Löf

引用次数: 7

Continuous Abstract Data Types for Verified Computation 用于验证计算的连续抽象数据类型

The Bulletin of Symbolic Logic Pub Date : 2021-12-01 DOI: 10.1017/bsl.2021.51

Sewon Park

{"title":"Continuous Abstract Data Types for Verified Computation","authors":"Sewon Park","doi":"10.1017/bsl.2021.51","DOIUrl":"https://doi.org/10.1017/bsl.2021.51","url":null,"abstract":"Abstract We devise imperative programming languages for verified real number computation where real numbers are provided as abstract data types such that the users of the languages can express real number computation by considering real numbers as abstract mathematical entities. Unlike other common approaches toward real number computation, based on an algebraic model that lacks implementability or transcendental computation, or finite-precision approximation such as using double precision computation that lacks a formal foundation, our languages are devised based on computable analysis, a foundation of rigorous computation over continuous data. Consequently, the users of the language can easily program real number computation and reason about the behaviours of their programs, relying on their mathematical knowledge of real numbers without worrying about artificial roundoff errors. As the languages are imperative, we adopt precondition–postcondition-style program specification and Hoare-style program verification methodologies. Consequently, the users of the language can easily program a computation over real numbers, specify the expected behaviour of the program, including termination, and prove the correctness of the specification. Furthermore, we suggest extending the languages with other interesting continuous data, such as matrices, continuous real functions, et cetera. Abstract taken directly from the thesis. E-mail: sewonpark17@gmail.com URL: https://sewonpark.com/thesis","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":"14 1","pages":"531 - 531"},"PeriodicalIF":0.0,"publicationDate":"2021-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79857815","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

SENSE AND REFERENCE FROM A CONSTRUCTIVIST STANDPOINT 从建构主义的角度看意义和指称

The Bulletin of Symbolic Logic Pub Date : 2021-11-15 DOI: 10.1017/bsl.2021.60

M. Dummett

引用次数: 1