The Bulletin of Symbolic Logic最新文献_第2页

EXTENDED FRAMES AND SEPARATIONS OF LOGICAL PRINCIPLES 扩展框架和逻辑原则的分离

The Bulletin of Symbolic Logic Pub Date : 2023-07-26 DOI: 10.1017/bsl.2023.29

M. Fujiwara, H. Ishihara, Takako Nemoto, Nobu-Yuki Suzuki, K. Yokoyama

引用次数: 2

Around Exponential-Algebraic Closedness 围绕指数代数封闭性

The Bulletin of Symbolic Logic Pub Date : 2023-06-01 DOI: 10.1017/bsl.2022.46

F. Gallinaro

{"title":"Around Exponential-Algebraic Closedness","authors":"F. Gallinaro","doi":"10.1017/bsl.2022.46","DOIUrl":"https://doi.org/10.1017/bsl.2022.46","url":null,"abstract":"Abstract We present some results related to Zilber’s Exponential-Algebraic Closedness Conjecture, showing that various systems of equations involving algebraic operations and certain analytic functions admit solutions in the complex numbers. These results are inspired by Zilber’s theorems on raising to powers. We show that algebraic varieties which split as a product of a linear subspace of an additive group and an algebraic subvariety of a multiplicative group intersect the graph of the exponential function, provided that they satisfy Zilber’s freeness and rotundity conditions, using techniques from tropical geometry. We then move on to prove a similar theorem, establishing that varieties which split as a product of a linear subspace and a subvariety of an abelian variety A intersect the graph of the exponential map of A (again under the analogues of the freeness and rotundity conditions). The proof uses homology and cohomology of manifolds. Finally, we show that the graph of the modular j-function intersects varieties which satisfy freeness and broadness and split as a product of a Möbius subvariety of a power of the upper-half plane and a complex algebraic variety, using Ratner’s orbit closure theorem to study the images under j of Möbius varieties. Abstract prepared by Francesco Paolo Gallinaro E-mail: francesco.gallinaro@mathematik.uni-freiburg.de URL: https://etheses.whiterose.ac.uk/31077/","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":"33 1","pages":"300 - 300"},"PeriodicalIF":0.0,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85558961","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

BSL volume 29 issue 2 Cover and Front matter BSL第29卷第2期封面和封面问题

The Bulletin of Symbolic Logic Pub Date : 2023-06-01 DOI: 10.1017/bsl.2023.20

G. Bezhanishvili, S. Kuhlmann, K. Bimbó, Øystein Linnebo, P. Dybjer, A. Muscholl, A. Enayat, Arno Pauly, Albert Atserias, Antonio Montalbán, M. Atten, V. D. Paiva, Clinton Conley, Christian Retoré, D. Macpherson, Nam Trang, Sandra Müller

引用次数: 0

Jan Krajìček, Proof Complexity, Encyclopedia of Mathematics and Its Applications, no. 170, Cambridge University Press, Cambridge, UK, 2019, xvi + 516 pp. Jan Krajìček，证明复杂性，数学百科全书及其应用，第2期。170，剑桥大学出版社，剑桥，英国，2019年，16 + 516页。

The Bulletin of Symbolic Logic Pub Date : 2023-06-01 DOI: 10.1017/bsl.2023.13

M. Müller

引用次数: 0

Effective Concept Classes of PAC and PACi Incomparable Degrees, Joins and Embedding of Degrees PAC和PACi的不可比度、连接和嵌入的有效概念类

The Bulletin of Symbolic Logic Pub Date : 2023-06-01 DOI: 10.1017/bsl.2022.39

D. G. M. Senadheera

{"title":"Effective Concept Classes of PAC and PACi Incomparable Degrees, Joins and Embedding of Degrees","authors":"D. G. M. Senadheera","doi":"10.1017/bsl.2022.39","DOIUrl":"https://doi.org/10.1017/bsl.2022.39","url":null,"abstract":"Abstract The Probably Approximately Correct (PAC) learning is a machine learning model introduced by Leslie Valiant in 1984. The PACi reducibility refers to the PAC reducibility independent of size and computation time. This reducibility in PAC learning resembles the reducibility in Turing computability. The ordering of concept classes under PAC reducibility is nonlinear, even when restricted to particular concrete examples. Due to the resemblance to Turing Reducibility, we suspected that there could be incomparable PACi and PAC degrees for the PACi and PAC reducibilities as in Turing incomparable degrees. In 1957 Friedberg and in 1956 Muchnik independently solved the Post problem by constructing computably enumerable sets A and B of incomparable degrees using the priority construction method. We adapt this idea to PACi and PAC reducibilities and construct two effective concept classes C and D such that C is not reducible to D and vice versa. When considering PAC reducibility it was necessary to work on the size of an effective concept class, thus we use Kolmogorov complexity to obtain the size. The non-learnability of concept classes in the PAC learning model is explained by the existence of PAC incomparable degrees. Analogous to the Turing jump, we give a jump operation on effective concept classes for the zero jump. To define the zero jump operator for PACi degrees the join of all the effective concept classes is constructed and proved that it is a greatest element. There are many properties proven for existing degrees. Thus we can explore proving those properties to PACi and PAC degrees. But if we prove an embedding from those degrees to PACi and PAC degrees then those properties will be true for PACi and PAC degrees without explicitly proving them. Abstract prepared by Dodamgodage Gihnee M. Senadheera and taken directly from the thesis E-mail: senadheerad@winthrop.edu URL: https://www.proquest.com/docview/2717762461/abstract/ACD19F29A8774AF6PQ/1?accountid=13864","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":"29 1","pages":"298 - 299"},"PeriodicalIF":0.0,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86744282","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

BSL volume 29 issue 2 Cover and Back matter BSL第29卷第2期封面和封底

The Bulletin of Symbolic Logic Pub Date : 2023-06-01 DOI: 10.1017/bsl.2023.21

引用次数: 0

New methods in forcing iteration and applications 强制迭代的新方法及应用

The Bulletin of Symbolic Logic Pub Date : 2023-06-01 DOI: 10.1017/bsl.2023.7

R. Mohammadpour

{"title":"New methods in forcing iteration and applications","authors":"R. Mohammadpour","doi":"10.1017/bsl.2023.7","DOIUrl":"https://doi.org/10.1017/bsl.2023.7","url":null,"abstract":"Abstract The Theme. Strong forcing axioms like Martin’s Maximum give a reasonably satisfactory structural analysis of \u0000$H(omega _2)$\u0000 . A broad program in modern Set Theory is searching for strong forcing axioms beyond \u0000$omega _1$\u0000 . In other words, one would like to figure out the structural properties of taller initial segments of the universe. However, the classical techniques of forcing iterations seem unable to bypass the obstacles, as the resulting forcings axioms beyond \u0000$omega _1$\u0000 have not thus far been strong enough! However, with his celebrated work on generalised side conditions, I. Neeman introduced us to a novel paradigm to iterate forcings. In particular, he could, among other things, reprove the consistency of the Proper Forcing Axiom using an iterated forcing with finite supports. In 2015, using his technology of virtual models, Veličković built up an iteration of semi-proper forcings with finite supports, hence reproving the consistency of Martin’s Maximum, an achievement leading to the notion of a virtual model. In this thesis, we are interested in constructing forcing notions with finitely many virtual models as side conditions to preserve three uncountable cardinals. The thesis constitutes six chapters and three appendices that amount to 118 pages, where Section 1 is devoted to preliminaries, and Section 2 is a warm-up about the scaffolding poset of a proper forcing. In Section 3, we present the general theory of virtual models in the context of forcing with sets of models of two types, where we, e.g., define the “meet” between two virtual models and prove its properties. The main results are joint with Boban Veličković, and partly appeared in Guessing models and the approachability ideal, J. Math. Log. 21 (2021). Pure Side Conditions. In Section 4, we use two types of virtual models (countable and large non-transitive ones induced by a supercompact cardinal, which we call Magidor models) to construct our forcing with pure side conditions. The forcing covertly uses a third type of models that are transitive. We also add decorations to the conditions to add many clubs in the generic \u0000$omega _2$\u0000 . In contrast to Neeman’s method, we do not have a single chain, but \u0000$alpha $\u0000 -chains, for an ordinal \u0000$alpha $\u0000 with \u0000$V_alpha prec V_lambda $\u0000 . Thus, starting from suitable large cardinals \u0000$kappa <lambda $\u0000 , we construct a forcing notion whose conditions are finite sets of virtual models described earlier. The forcing is strongly proper, preserves \u0000$kappa $\u0000 , and has the \u0000$lambda $\u0000 -Knaster property. The relevant quotients of the forcing are strongly proper, which helps us prove strong guessing model principles. The construction is generalisable to a \u0000${<}mu $\u0000 -closed forcing, for any given cardinal \u0000$mu $\u0000 with \u0000$mu ^{<mu }=mu <kappa $\u0000 . The Iteration Theorem. In Section 5, we use the forcing with pure side conditions to iterate a subclass of proper and \u0000$aleph _2$\u0000 -c.c. forcings and obtain a forcing axiom at the level of ","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":"39 1","pages":"300 - 302"},"PeriodicalIF":0.0,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77728237","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: 4

Katie Steele and H. Orri Stefánsson. Beyond Uncertainty: Reasoning with Unknown Possibilities. Elements in Decision Theory and Philosophy. Cambridge University Press, Cambridge, UK, 2021, 110 pp. Katie Steele和H. Orri Stefánsson。超越不确定性:用未知的可能性推理。决策理论与哲学的要素。剑桥大学出版社，剑桥，英国，2021年，110页。

The Bulletin of Symbolic Logic Pub Date : 2023-06-01 DOI: 10.1017/bsl.2023.11

Magdalen Elmitt

引用次数: 0

Forcing theory and combinatorics of the real line 实线的强迫理论与组合

The Bulletin of Symbolic Logic Pub Date : 2023-06-01 DOI: 10.1017/bsl.2022.40

Miguel Antonio Cardona-Montoya

{"title":"Forcing theory and combinatorics of the real line","authors":"Miguel Antonio Cardona-Montoya","doi":"10.1017/bsl.2022.40","DOIUrl":"https://doi.org/10.1017/bsl.2022.40","url":null,"abstract":"Abstract The main purpose of this dissertation is to apply and develop new forcing techniques to obtain models where several cardinal characteristics are pairwise different as well as force many (even more, continuum many) different values of cardinal characteristics that are parametrized by reals. In particular, we look at cardinal characteristics associated with strong measure zero, Yorioka ideals, and localization and anti-localization cardinals. In this thesis we introduce the property “F-linked” of subsets of posets for a given free filter F on the natural numbers, and define the properties “ \u0000$mu $\u0000 -F-linked” and “ \u0000$theta $\u0000 -F-Knaster” for posets in a natural way. We show that \u0000$theta $\u0000 -F-Knaster posets preserve strong types of unbounded families and of maximal almost disjoint families. These kinds of posets led to the development of a general technique to construct \u0000$theta $\u0000 - \u0000$textrm {Fr}$\u0000 -Knaster posets (where \u0000$textrm {Fr}$\u0000 is the Frechet ideal) via matrix iterations of \u0000${<}theta $\u0000 -ultrafilter-linked posets (restricted to some level of the matrix). The latter technique allows proving consistency results about Cichoń’s diagram (without using large cardinals) and to prove the consistency of the fact that, for each Yorioka ideal, the four cardinal characteristics associated with it are pairwise different. Another important application is to show that three strongly compact cardinals are enough to force that Cichoń’s diagram can be separated into 10 different values. Later on, it was shown by Goldstern, Kellner, Mejía, and Shelah that no large cardinals are needed for Cichoń’s maximum (J. Eur. Math. Soc. 24 (2022), no. 11, p. 3951–3967). On the other hand, we deal with certain types of tree forcings including Sacks forcing, and show that these increase the covering of the strong measure zero ideal \u0000$mathcal {SN}$\u0000 . As a consequence, in Sacks model, such covering number is equal to the size of the continuum, which indicates that this covering number is consistently larger than any other classical cardinal characteristics of the continuum. Even more, Sacks forcing can be used to force that \u0000$operatorname {mathrm{non}}(mathcal {SN})<operatorname {mathrm{cov}}(mathcal {SN})<operatorname {mathrm{cof}}(mathcal {SN})$\u0000 , which is the first consistency result where more than two cardinal characteristics associated with \u0000$mathcal {SN}$\u0000 are pairwise different. To obtain another result in this direction, we provide bounds for \u0000$operatorname {mathrm{cof}}(mathcal {SN})$\u0000 , which generalizes Yorioka’s characterization of \u0000$mathcal {SN}$\u0000 (J. Symbolic Logic 67.4 (2002), p. 1373–1384). As a consequence, we get the consistency of \u0000$operatorname {mathrm{add}}(mathcal {SN})=operatorname {mathrm{cov}}(mathcal {SN})<operatorname {mathrm{non}}(mathcal {SN})<operatorname {mathrm{cof}}(mathcal {SN})$\u0000 with ZFC (via a matrix iteration forcing construction). We conclude this thesis by combining creature forcing approaches by Kellner and Shelah (Arch.","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":"22 1","pages":"299 - 300"},"PeriodicalIF":0.0,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75250567","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

COMPUTABLY COMPACT METRIC SPACES 可计算紧化度量空间

The Bulletin of Symbolic Logic Pub Date : 2023-05-11 DOI: 10.1017/bsl.2023.16

R. Downey, A. Melnikov

引用次数: 3