Pub Date : 2019-12-06DOI: 10.1142/S0219061321500197
Alejandro Poveda, A. Rinot, Dima Sinapova
In Part I of this series [5], we introduced a class of notions of forcing which we call [Formula: see text]-Prikry, and showed that many of the known Prikry-type notions of forcing that centers around singular cardinals of countable cofinality are [Formula: see text]-Prikry. We proved that given a [Formula: see text]-Prikry poset [Formula: see text] and a [Formula: see text]-name for a nonreflecting stationary set [Formula: see text], there exists a corresponding [Formula: see text]-Prikry poset that projects to [Formula: see text] and kills the stationarity of [Formula: see text]. In this paper, we develop a general scheme for iterating [Formula: see text]-Prikry posets, as well as verify that the Extender-based Prikry forcing is [Formula: see text]-Prikry. As an application, we blow-up the power of a countable limit of Laver-indestructible supercompact cardinals, and then iteratively kill all nonreflecting stationary subsets of its successor. This yields a model in which the singular cardinal hypothesis fails and simultaneous reflection of finite families of stationary sets holds.
{"title":"Sigma-Prikry forcing II: Iteration Scheme","authors":"Alejandro Poveda, A. Rinot, Dima Sinapova","doi":"10.1142/S0219061321500197","DOIUrl":"https://doi.org/10.1142/S0219061321500197","url":null,"abstract":"In Part I of this series [5], we introduced a class of notions of forcing which we call [Formula: see text]-Prikry, and showed that many of the known Prikry-type notions of forcing that centers around singular cardinals of countable cofinality are [Formula: see text]-Prikry. We proved that given a [Formula: see text]-Prikry poset [Formula: see text] and a [Formula: see text]-name for a nonreflecting stationary set [Formula: see text], there exists a corresponding [Formula: see text]-Prikry poset that projects to [Formula: see text] and kills the stationarity of [Formula: see text]. In this paper, we develop a general scheme for iterating [Formula: see text]-Prikry posets, as well as verify that the Extender-based Prikry forcing is [Formula: see text]-Prikry. As an application, we blow-up the power of a countable limit of Laver-indestructible supercompact cardinals, and then iteratively kill all nonreflecting stationary subsets of its successor. This yields a model in which the singular cardinal hypothesis fails and simultaneous reflection of finite families of stationary sets holds.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":"30 1","pages":"2150019:1-2150019:59"},"PeriodicalIF":0.9,"publicationDate":"2019-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73078895","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-12-06DOI: 10.1142/s0219061321500021
C. Lambie-Hanson, A. Rinot
Motivated by a characterization of weakly compact cardinals due to Todorcevic, we introduce a new cardinal characteristic, the [Formula: see text]-sequence number, which can be seen as a measure of the compactness of a regular uncountable cardinal. We prove a number of [Formula: see text] and independence results about the [Formula: see text]-sequence number and its relationship with large cardinals, stationary reflection, and square principles. We then introduce and study the more general [Formula: see text]-sequence spectrum and uncover some tight connections between the [Formula: see text]-sequence spectrum and the strong coloring principle [Formula: see text], introduced in Part I of this series.
{"title":"Knaster and friends II: The C-sequence number","authors":"C. Lambie-Hanson, A. Rinot","doi":"10.1142/s0219061321500021","DOIUrl":"https://doi.org/10.1142/s0219061321500021","url":null,"abstract":"Motivated by a characterization of weakly compact cardinals due to Todorcevic, we introduce a new cardinal characteristic, the [Formula: see text]-sequence number, which can be seen as a measure of the compactness of a regular uncountable cardinal. We prove a number of [Formula: see text] and independence results about the [Formula: see text]-sequence number and its relationship with large cardinals, stationary reflection, and square principles. We then introduce and study the more general [Formula: see text]-sequence spectrum and uncover some tight connections between the [Formula: see text]-sequence spectrum and the strong coloring principle [Formula: see text], introduced in Part I of this series.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":"33 1","pages":"2150002:1-2150002:54"},"PeriodicalIF":0.9,"publicationDate":"2019-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74292955","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-10-02DOI: 10.1142/S0219061319500089
Ashutosh Kumar, S. Shelah
We prove that the null ideal restricted to a non-null set of reals could be isomorphic to a variety of sigma ideals. Using this, we show that the following are consistent: (1) There is a non-null subset of plane each of whose non-null subsets contains three collinear points. (2) There is a partition of a non-null set of reals into null sets, each of size [Formula: see text], such that every transversal of this partition is null.
{"title":"On possible restrictions of the null ideal","authors":"Ashutosh Kumar, S. Shelah","doi":"10.1142/S0219061319500089","DOIUrl":"https://doi.org/10.1142/S0219061319500089","url":null,"abstract":"We prove that the null ideal restricted to a non-null set of reals could be isomorphic to a variety of sigma ideals. Using this, we show that the following are consistent: (1) There is a non-null subset of plane each of whose non-null subsets contains three collinear points. (2) There is a partition of a non-null set of reals into null sets, each of size [Formula: see text], such that every transversal of this partition is null.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":"55 1","pages":"1950008:1-1950008:14"},"PeriodicalIF":0.9,"publicationDate":"2019-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80431170","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-09-17DOI: 10.1142/s0219061320500257
Francesco Mangraviti, L. Ros
Answering one of the main questions of [S.-D. Friedman, T. Hyttinen and V. Kulikov, Generalized descriptive set theory and classification theory, Mem. Amer. Math. Soc. 230(1081) (2014) 80, Chap. 7], we show that there is a tight connection between the depth of a classifiable shallow theory [Formula: see text] and the Borel rank of the isomorphism relation [Formula: see text] on its models of size [Formula: see text], for [Formula: see text] any cardinal satisfying [Formula: see text]. This is achieved by establishing a link between said rank and the [Formula: see text]-Scott height of the [Formula: see text]-sized models of [Formula: see text], and yields to the following descriptive set-theoretical analog of Shelah’s Main Gap Theorem: Given a countable complete first-order theory [Formula: see text], either [Formula: see text] is Borel with a countable Borel rank (i.e. very simple, given that the length of the relevant Borel hierarchy is [Formula: see text]), or it is not Borel at all. The dividing line between the two situations is the same as in Shelah’s theorem, namely that of classifiable shallow theories. We also provide a Borel reducibility version of the above theorem, discuss some limitations to the possible (Borel) complexities of [Formula: see text], and provide a characterization of categoricity of [Formula: see text] in terms of the descriptive set-theoretical complexity of [Formula: see text].
回答[s . d .]傅利民，《广义描述集理论与分类理论》，《中国科学》。阿米尔。数学。Soc. 230(1081)(2014) 80，第7章)，我们证明了在可分类的浅层理论[公式:见文]的深度与其大小[公式:见文]模型上同构关系[公式:见文]的Borel秩之间存在紧密联系，对于[公式:见文]任何满足[公式:见文]的基数。这是通过在所述秩和[公式:见文]-[公式:见文]的[公式:见文]-大小模型的[公式:见文]之间建立联系来实现的，并产生以下描述集理论类似于Shelah的主要间隙定理:给定一个可数的完全一阶理论[公式:见文]，要么[公式:见文]是具有可数Borel秩的Borel(即非常简单，给定相关Borel层次的长度为[公式:(见原文)，或者根本就不是Borel。这两种情况之间的分界线与希拉定理相同，即可分类的浅层理论的分界线。我们还提供了上述定理的Borel可约性版本，讨论了[公式:见文]的可能(Borel)复杂性的一些限制，并根据[公式:见文]的描述性集合理论复杂性提供了[公式:见文]的范畴性的表征。
{"title":"A descriptive Main Gap Theorem","authors":"Francesco Mangraviti, L. Ros","doi":"10.1142/s0219061320500257","DOIUrl":"https://doi.org/10.1142/s0219061320500257","url":null,"abstract":"Answering one of the main questions of [S.-D. Friedman, T. Hyttinen and V. Kulikov, Generalized descriptive set theory and classification theory, Mem. Amer. Math. Soc. 230(1081) (2014) 80, Chap. 7], we show that there is a tight connection between the depth of a classifiable shallow theory [Formula: see text] and the Borel rank of the isomorphism relation [Formula: see text] on its models of size [Formula: see text], for [Formula: see text] any cardinal satisfying [Formula: see text]. This is achieved by establishing a link between said rank and the [Formula: see text]-Scott height of the [Formula: see text]-sized models of [Formula: see text], and yields to the following descriptive set-theoretical analog of Shelah’s Main Gap Theorem: Given a countable complete first-order theory [Formula: see text], either [Formula: see text] is Borel with a countable Borel rank (i.e. very simple, given that the length of the relevant Borel hierarchy is [Formula: see text]), or it is not Borel at all. The dividing line between the two situations is the same as in Shelah’s theorem, namely that of classifiable shallow theories. We also provide a Borel reducibility version of the above theorem, discuss some limitations to the possible (Borel) complexities of [Formula: see text], and provide a characterization of categoricity of [Formula: see text] in terms of the descriptive set-theoretical complexity of [Formula: see text].","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":"25 1","pages":"2050025:1-2050025:40"},"PeriodicalIF":0.9,"publicationDate":"2019-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77037323","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-06-11DOI: 10.1142/s021906132150001x
William Chan
If [Formula: see text] is a proper Polish metric space and [Formula: see text] is any countable dense submetric space of [Formula: see text], then the Scott rank of [Formula: see text] in the natural first-order language of metric spaces is countable and in fact at most [Formula: see text], where [Formula: see text] is the Church–Kleene ordinal of [Formula: see text] (construed as a subset of [Formula: see text]) which is the least ordinal with no presentation on [Formula: see text] computable from [Formula: see text]. If [Formula: see text] is a rigid Polish metric space and [Formula: see text] is any countable dense submetric space, then the Scott rank of [Formula: see text] is countable and in fact less than [Formula: see text].
如果[公式:见文]是一个适当的波兰度量空间，而[公式:见文]是[公式:见文]的任何可数稠密子度量空间，那么[公式:见文]在度量空间的自然一阶语言中的Scott秩是可数的，实际上至多[公式:见文]，其中[公式:见文]是[公式:见文]的Church-Kleene序数(解释为[公式:见文]的一个子集)，它是没有表示的最小序数[公式:见文]:可由[公式:见文本]计算的。如果[Formula: see text]是一个严格的波兰度量空间，而[Formula: see text]是任何可数的密集子度量空间，那么[Formula: see text]的Scott秩是可数的，实际上小于[Formula: see text]。
{"title":"Bounds on Scott ranks of some polish metric spaces","authors":"William Chan","doi":"10.1142/s021906132150001x","DOIUrl":"https://doi.org/10.1142/s021906132150001x","url":null,"abstract":"If [Formula: see text] is a proper Polish metric space and [Formula: see text] is any countable dense submetric space of [Formula: see text], then the Scott rank of [Formula: see text] in the natural first-order language of metric spaces is countable and in fact at most [Formula: see text], where [Formula: see text] is the Church–Kleene ordinal of [Formula: see text] (construed as a subset of [Formula: see text]) which is the least ordinal with no presentation on [Formula: see text] computable from [Formula: see text]. If [Formula: see text] is a rigid Polish metric space and [Formula: see text] is any countable dense submetric space, then the Scott rank of [Formula: see text] is countable and in fact less than [Formula: see text].","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":"108 1","pages":"2150001:1-2150001:23"},"PeriodicalIF":0.9,"publicationDate":"2019-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83349892","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-05-23DOI: 10.1142/s0219061322500301
D. Hoffmann, Junguk Lee
We achieve several results. First, we develop a variant of the theory of absolute Galois groups in the context of many sorted structures. Second, we provide a method for coding absolute Galois groups of structures, so they can be interpreted in some monster model with an additional predicate. Third, we prove a "weak independence theorem" for PAC substructures of an ambient structure with nfcp and property B(3). Fourth, we describe Kim-dividing in these PAC substructures and show several results related to NSOP. Fifth, we characterize the algebraic closure in PAC structures.
{"title":"Co-theory of sorted profinite groups for PAC structures","authors":"D. Hoffmann, Junguk Lee","doi":"10.1142/s0219061322500301","DOIUrl":"https://doi.org/10.1142/s0219061322500301","url":null,"abstract":"We achieve several results. First, we develop a variant of the theory of absolute Galois groups in the context of many sorted structures. Second, we provide a method for coding absolute Galois groups of structures, so they can be interpreted in some monster model with an additional predicate. Third, we prove a \"weak independence theorem\" for PAC substructures of an ambient structure with nfcp and property B(3). Fourth, we describe Kim-dividing in these PAC substructures and show several results related to NSOP. Fifth, we characterize the algebraic closure in PAC structures.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":"35 1","pages":"2250030:1-2250030:60"},"PeriodicalIF":0.9,"publicationDate":"2019-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77095562","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-05-21DOI: 10.1142/S0219061321500136
B. Monin, Ludovic Patey
The infinite pigeonhole principle for 2-partitions ([Formula: see text]) asserts the existence, for every set [Formula: see text], of an infinite subset of [Formula: see text] or of its complement. In this paper, we study the infinite pigeonhole principle from a computability-theoretic viewpoint. We prove in particular that [Formula: see text] admits strong cone avoidance for arithmetical and hyperarithmetical reductions. We also prove the existence, for every [Formula: see text] set, of an infinite low[Formula: see text] subset of it or its complement. This answers a question of Wang. For this, we design a new notion of forcing which generalizes the first and second-jump control of Cholak et al.
{"title":"The weakness of the pigeonhole principle under hyperarithmetical reductions","authors":"B. Monin, Ludovic Patey","doi":"10.1142/S0219061321500136","DOIUrl":"https://doi.org/10.1142/S0219061321500136","url":null,"abstract":"The infinite pigeonhole principle for 2-partitions ([Formula: see text]) asserts the existence, for every set [Formula: see text], of an infinite subset of [Formula: see text] or of its complement. In this paper, we study the infinite pigeonhole principle from a computability-theoretic viewpoint. We prove in particular that [Formula: see text] admits strong cone avoidance for arithmetical and hyperarithmetical reductions. We also prove the existence, for every [Formula: see text] set, of an infinite low[Formula: see text] subset of it or its complement. This answers a question of Wang. For this, we design a new notion of forcing which generalizes the first and second-jump control of Cholak et al.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":"10 1","pages":"2150013:1-2150013:41"},"PeriodicalIF":0.9,"publicationDate":"2019-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78839612","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-05-20DOI: 10.1142/s0219061320500166
Daniel Turetsky
Using new techniques for controlling the categoricity spectrum of a structure, we construct a structure with degree of categoricity but infinite spectral dimension, answering a question of Bazhenov, Kalimullin and Yamaleev. Using the same techniques, we construct a computably categorical structure of non-computable Scott rank.
{"title":"Coding in the automorphism group of a computably categorical structure","authors":"Daniel Turetsky","doi":"10.1142/s0219061320500166","DOIUrl":"https://doi.org/10.1142/s0219061320500166","url":null,"abstract":"Using new techniques for controlling the categoricity spectrum of a structure, we construct a structure with degree of categoricity but infinite spectral dimension, answering a question of Bazhenov, Kalimullin and Yamaleev. Using the same techniques, we construct a computably categorical structure of non-computable Scott rank.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":"41 1","pages":"2050016:1-2050016:24"},"PeriodicalIF":0.9,"publicationDate":"2019-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86390027","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-05-17DOI: 10.1142/s0219061321500070
A. Fornasiero, E. Kaplan
Let $T$ be a complete, model complete o-minimal theory extending the theory RCF of real closed ordered fields in some appropriate language $L$. We study derivations $delta$ on models $mathcal{M}models T$. We introduce the notion of a $T$-derivation: a derivation which is compatible with the $L(emptyset)$-definable $mathcal{C}^1$-functions on $mathcal{M}$. We show that the theory of $T$-models with a $T$-derivation has a model completion $T^delta_G$. The derivation in models $(mathcal{M},delta)models T^delta_G$ behaves "generically," it is wildly discontinuous and its kernel is a dense elementary $L$-substructure of $mathcal{M}$. If $T =$ RCF, then $T^delta_G$ is the theory of closed ordered differential fields (CODF) as introduced by Michael Singer. We are able to recover many of the known facts about CODF in our setting. Among other things, we show that $T^delta_G$ has $T$ as its open core, that $T^delta_G$ is distal, and that $T^delta_G$ eliminates imaginaries. We also show that the theory of $T$-models with finitely many commuting $T$-derivations has a model completion.
{"title":"Generic derivations on o-minimal structures","authors":"A. Fornasiero, E. Kaplan","doi":"10.1142/s0219061321500070","DOIUrl":"https://doi.org/10.1142/s0219061321500070","url":null,"abstract":"Let $T$ be a complete, model complete o-minimal theory extending the theory RCF of real closed ordered fields in some appropriate language $L$. We study derivations $delta$ on models $mathcal{M}models T$. We introduce the notion of a $T$-derivation: a derivation which is compatible with the $L(emptyset)$-definable $mathcal{C}^1$-functions on $mathcal{M}$. We show that the theory of $T$-models with a $T$-derivation has a model completion $T^delta_G$. The derivation in models $(mathcal{M},delta)models T^delta_G$ behaves \"generically,\" it is wildly discontinuous and its kernel is a dense elementary $L$-substructure of $mathcal{M}$. If $T =$ RCF, then $T^delta_G$ is the theory of closed ordered differential fields (CODF) as introduced by Michael Singer. We are able to recover many of the known facts about CODF in our setting. Among other things, we show that $T^delta_G$ has $T$ as its open core, that $T^delta_G$ is distal, and that $T^delta_G$ eliminates imaginaries. We also show that the theory of $T$-models with finitely many commuting $T$-derivations has a model completion.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":"55 1","pages":"2150007:1-2150007:45"},"PeriodicalIF":0.9,"publicationDate":"2019-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76135318","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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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