M. Fujiwara, H. Ishihara, Takako Nemoto, Nobu-Yuki Suzuki, K. Yokoyama
Abstract We aim at developing a systematic method of separating omniscience principles by constructing Kripke models for intuitionistic predicate logic $mathbf {IQC}$ and first-order arithmetic $mathbf {HA}$ from a Kripke model for intuitionistic propositional logic $mathbf {IPC}$ . To this end, we introduce the notion of an extended frame, and show that each IPC-Kripke model generates an extended frame. By using the extended frame generated by an IPC-Kripke model, we give a separation theorem of a schema from a set of schemata in $mathbf {IQC}$ and a separation theorem of a sentence from a set of schemata in $mathbf {HA}$ . We see several examples which give us separations among omniscience principles.
{"title":"EXTENDED FRAMES AND SEPARATIONS OF LOGICAL PRINCIPLES","authors":"M. Fujiwara, H. Ishihara, Takako Nemoto, Nobu-Yuki Suzuki, K. Yokoyama","doi":"10.1017/bsl.2023.29","DOIUrl":"https://doi.org/10.1017/bsl.2023.29","url":null,"abstract":"Abstract We aim at developing a systematic method of separating omniscience principles by constructing Kripke models for intuitionistic predicate logic \u0000$mathbf {IQC}$\u0000 and first-order arithmetic \u0000$mathbf {HA}$\u0000 from a Kripke model for intuitionistic propositional logic \u0000$mathbf {IPC}$\u0000 . To this end, we introduce the notion of an extended frame, and show that each IPC-Kripke model generates an extended frame. By using the extended frame generated by an IPC-Kripke model, we give a separation theorem of a schema from a set of schemata in \u0000$mathbf {IQC}$\u0000 and a separation theorem of a sentence from a set of schemata in \u0000$mathbf {HA}$\u0000 . We see several examples which give us separations among omniscience principles.","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":"10 1","pages":"311 - 353"},"PeriodicalIF":0.0,"publicationDate":"2023-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76700579","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}
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/
摘要本文给出了关于Zilber指数-代数闭性猜想的一些结果，证明了各种涉及代数运算的方程组和某些解析函数在复数中有解。这些结果的灵感来自于齐尔伯关于幂次幂的定理。我们使用热带几何的技术证明了作为可加群的线性子空间和乘法群的代数子空间的乘积分裂的代数变体相交于指数函数的图，只要它们满足Zilber的自由和圆度条件。然后我们继续证明一个类似的定理，建立作为线性子空间和阿贝尔变体a的子变体的乘积分裂的变体与a的指数映射的图相交(再次在自由和圆度条件的类似情况下)。证明使用流形的同调和上同调。最后，我们利用Ratner的轨道闭包定理研究了Möbius种j下的像，证明了模j函数的图与满足自由度和宽度的变种相交并分裂为上半平面幂次的Möbius子变种与复代数变种的乘积。作者:Francesco Paolo Gallinaro E-mail: francesco.gallinaro@mathematik.uni-freiburg.de URL: https://etheses.whiterose.ac.uk/31077/
{"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}
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
and
和
{"title":"BSL volume 29 issue 2 Cover and Front matter","authors":"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","doi":"10.1017/bsl.2023.20","DOIUrl":"https://doi.org/10.1017/bsl.2023.20","url":null,"abstract":"and","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":"92 3","pages":"f1 - f3"},"PeriodicalIF":0.0,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91483708","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}
and ‘??’ is a subjective catch-all standing in for those contingencies that she suspects she is unaware of. Represented like this, Steele and Stefánsson argue that the policy-maker’s predicament is somewhat unremarkable and (if certain basic conditions are met) we can treat her as an EU maximiser just like any ordinary reasoner. However, they go on to canvas two norms of rationality—‘Awareness Reflection’ and ‘Preference Awareness Reflection’—which they think should constrain the synchronic credences and desires (respectively) of agents like the policy-maker who anticipate their awareness will grow in rather specific ways. Whilst the positive proposal spelled out in Sections 6 and 7 leaves several questions open, Steele and Stefánsson successfully lay the foundations for others working within normative decision theory and related areas of economics and computer science to take up these questions and continue the work of characterising the reasoning of rational, but less-than-fully aware, agents.
{"title":"Jan Krajìček, Proof Complexity, Encyclopedia of Mathematics and Its Applications, no. 170, Cambridge University Press, Cambridge, UK, 2019, xvi + 516 pp.","authors":"M. Müller","doi":"10.1017/bsl.2023.13","DOIUrl":"https://doi.org/10.1017/bsl.2023.13","url":null,"abstract":"and ‘??’ is a subjective catch-all standing in for those contingencies that she suspects she is unaware of. Represented like this, Steele and Stefánsson argue that the policy-maker’s predicament is somewhat unremarkable and (if certain basic conditions are met) we can treat her as an EU maximiser just like any ordinary reasoner. However, they go on to canvas two norms of rationality—‘Awareness Reflection’ and ‘Preference Awareness Reflection’—which they think should constrain the synchronic credences and desires (respectively) of agents like the policy-maker who anticipate their awareness will grow in rather specific ways. Whilst the positive proposal spelled out in Sections 6 and 7 leaves several questions open, Steele and Stefánsson successfully lay the foundations for others working within normative decision theory and related areas of economics and computer science to take up these questions and continue the work of characterising the reasoning of rational, but less-than-fully aware, agents.","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":"41 1","pages":"296 - 297"},"PeriodicalIF":0.0,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90208936","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}
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
大概近似正确(PAC)学习是Leslie Valiant在1984年提出的一种机器学习模型。PACi可约性是指与大小和计算时间无关的PACi可约性。PAC学习中的这种可约性类似于图灵可计算性中的可约性。在PAC可约性下，概念类的排序是非线性的，即使在特定的具体例子中也是如此。由于与图灵可约性的相似性，我们怀疑在图灵不可比拟度中，PACi和PAC可约性可能存在不可比拟的PACi和PAC度。1957年Friedberg和1956年Muchnik分别用优先级构造法构造了不可比较度的可计算枚举集合A和B，解决了Post问题。我们将这一思想应用于PACi和PAC可约性，并构造了两个有效的概念类C和D，使得C不可约为D，反之亦然。在考虑PAC可约性时，有必要研究有效概念类的大小，因此我们使用Kolmogorov复杂度来获得大小。PAC学习模型中概念类的不可学习性可以用PAC不可比较度的存在来解释。与图灵跳迁类似，我们给出了零跳迁的有效概念类的跳迁操作。为了定义PACi度的跳零算子，构造了所有有效概念类的联接，并证明了它是最大元。已有的学位已经证明了许多特性。因此，我们可以探索证明这些性质的PACi和PAC度。但是如果我们证明从这些度到PACi和PAC度的嵌入，那么这些属性将对PACi和PAC度成立，而不需要明确地证明它们。摘要由Dodamgodage Gihnee M. Senadheera撰写，直接摘自论文E-mail: senadheerad@winthrop.edu URL: https://www.proquest.com/docview/2717762461/abstract/ACD19F29A8774AF6PQ/1?accountid=13864
{"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}
Abstract The Theme. Strong forcing axioms like Martin’s Maximum give a reasonably satisfactory structural analysis of $H(omega _2)$ . A broad program in modern Set Theory is searching for strong forcing axioms beyond $omega _1$ . 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 $omega _1$ 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 $omega _2$ . In contrast to Neeman’s method, we do not have a single chain, but $alpha $ -chains, for an ordinal $alpha $ with $V_alpha prec V_lambda $ . Thus, starting from suitable large cardinals $kappa
{"title":"New methods in forcing iteration and applications","authors":"R. Mohammadpour","doi":"10.1017/bsl.2023.7","DOIUrl":"https://doi.org/10.1017/bsl.2023.7","url":null,"abstract":"Abstract The Theme. Strong forcing axioms like Martin’s Maximum give a reasonably satisfactory structural analysis of \u0000$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}
The Association for Symbolic Logic publishes analytical reviews of selected books and articles in the field of symbolic logic. The reviews were published in The Journal of Symbolic Logic from the founding of the journal in 1936 until the end of 1999. The Association moved the reviews to this bulletin, beginning in 2000. The Reviews Section is edited by Graham Leach-Krouse (Managing Editor), Albert Atserias, Mark van Atten, Clinton Conley, Johanna Franklin, Dugald Macpherson, Antonio Montalbán, Valeria de Paiva, Christian Retoré, Marion Scheepers, and Nam Trang. Authors and publishers are requested to send, for review, copies of books to ASL, Department of Mathematics, University of Connecticut, 341 Mansfield Road, U-1009, Storrs, CT 06269-1009, USA.
{"title":"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.","authors":"Magdalen Elmitt","doi":"10.1017/bsl.2023.11","DOIUrl":"https://doi.org/10.1017/bsl.2023.11","url":null,"abstract":"The Association for Symbolic Logic publishes analytical reviews of selected books and articles in the field of symbolic logic. The reviews were published in The Journal of Symbolic Logic from the founding of the journal in 1936 until the end of 1999. The Association moved the reviews to this bulletin, beginning in 2000. The Reviews Section is edited by Graham Leach-Krouse (Managing Editor), Albert Atserias, Mark van Atten, Clinton Conley, Johanna Franklin, Dugald Macpherson, Antonio Montalbán, Valeria de Paiva, Christian Retoré, Marion Scheepers, and Nam Trang. Authors and publishers are requested to send, for review, copies of books to ASL, Department of Mathematics, University of Connecticut, 341 Mansfield Road, U-1009, Storrs, CT 06269-1009, USA.","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":"8 1","pages":"294 - 296"},"PeriodicalIF":0.0,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73033240","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}
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 “ $mu $ -F-linked” and “ $theta $ -F-Knaster” for posets in a natural way. We show that $theta $ -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 $theta $ - $textrm {Fr}$ -Knaster posets (where $textrm {Fr}$ is the Frechet ideal) via matrix iterations of ${<}theta $ -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 $mathcal {SN}$ . 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 $operatorname {mathrm{non}}(mathcal {SN})
{"title":"Forcing theory and combinatorics of the real line","authors":"Miguel Antonio Cardona-Montoya","doi":"10.1017/bsl.2022.40","DOIUrl":"https://doi.org/10.1017/bsl.2022.40","url":null,"abstract":"Abstract The main purpose of this dissertation is to apply and develop new forcing techniques to obtain models where several cardinal characteristics are pairwise different as well as force many (even more, continuum many) different values of cardinal characteristics that are parametrized by reals. In particular, we look at cardinal characteristics associated with strong measure zero, Yorioka ideals, and localization and anti-localization cardinals. In this thesis we introduce the property “F-linked” of subsets of posets for a given free filter F on the natural numbers, and define the properties “ \u0000$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}
Abstract We give a systematic technical exposition of the foundations of the theory of computably compact metric spaces. We discover several new characterizations of computable compactness and apply these characterizations to prove new results in computable analysis and effective topology. We also apply the technique of computable compactness to give new and less combinatorially involved proofs of known results from the literature. Some of these results do not have computable compactness or compact spaces in their statements, and thus these applications are not necessarily direct or expected.
{"title":"COMPUTABLY COMPACT METRIC SPACES","authors":"R. Downey, A. Melnikov","doi":"10.1017/bsl.2023.16","DOIUrl":"https://doi.org/10.1017/bsl.2023.16","url":null,"abstract":"Abstract We give a systematic technical exposition of the foundations of the theory of computably compact metric spaces. We discover several new characterizations of computable compactness and apply these characterizations to prove new results in computable analysis and effective topology. We also apply the technique of computable compactness to give new and less combinatorially involved proofs of known results from the literature. Some of these results do not have computable compactness or compact spaces in their statements, and thus these applications are not necessarily direct or expected.","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":"12 1","pages":"170 - 263"},"PeriodicalIF":0.0,"publicationDate":"2023-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81555206","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}
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