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Invariant measures in simple and in small theories 简单理论和小理论中的不变测度
IF 0.9 1区 数学
Journal of Mathematical Logic Pub Date : 2021-05-15 DOI: 10.1142/s0219061322500258
A. Chernikov, E. Hrushovski, Alex Kruckman, K. Krupiński, Slavko Moconja, A. Pillay, N. Ramsey
{"title":"Invariant measures in simple and in small theories","authors":"A. Chernikov, E. Hrushovski, Alex Kruckman, K. Krupiński, Slavko Moconja, A. Pillay, N. Ramsey","doi":"10.1142/s0219061322500258","DOIUrl":"https://doi.org/10.1142/s0219061322500258","url":null,"abstract":"We give examples of (i) a simple theory with a formula (with parameters) which does not fork over the empty set but has mu measure 0 for every automorphism invariant Keisler measure mu, and (ii) a definable group G in a simple theory such that G is not definably amenable, i.e. there is no translation invariant Keisler measure on G We also discuss paradoxical decompositions both in the setting of discrete groups and of definable groups, and prove some positive results about small theories, including the definable amenability of definable groups, and nontriviality of the graded Grothendieck ring.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":"320 1","pages":"2250025:1-2250025:37"},"PeriodicalIF":0.9,"publicationDate":"2021-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76287633","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}
引用次数: 3
Actions of tame abelian product groups 驯服阿贝尔产品组的行为
IF 0.9 1区 数学
Journal of Mathematical Logic Pub Date : 2021-05-11 DOI: 10.1142/S0219061322500283
Shaun Allison, Assaf Shani
{"title":"Actions of tame abelian product groups","authors":"Shaun Allison, Assaf Shani","doi":"10.1142/S0219061322500283","DOIUrl":"https://doi.org/10.1142/S0219061322500283","url":null,"abstract":"A Polish group $G$ is tame if for any continuous action of $G$, the corresponding orbit equivalence relation is Borel. When $G = prod_n Gamma_n$ for countable abelian $Gamma_n$, Solecki (1995) gave a characterization for when $G$ is tame. Ding and Gao (2017) showed that for such $G$, the orbit equivalence relation must in fact be potentially $mathbf{Pi}^0_6$, while conjecturing that the optimal bound could be $mathbf{Pi}^0_3$. We show that the optimal bound is $D(mathbf{Pi}^0_5)$ by constructing an action of such a group $G$ which is not potentially $mathbf{Pi}^0_5$, and show how to modify the analysis of Ding and Gao to get this slightly better upper bound. It follows, using the results of Hjorth, Kechris, and Louvaeu (1998), that this is the optimal bound for the potential complexity of actions of tame abelian product groups. Our lower-bound analysis involves forcing over models of set theory where choice fails for sequences of finite sets.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":"29 1","pages":"2250028:1-2250028:25"},"PeriodicalIF":0.9,"publicationDate":"2021-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73624844","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}
引用次数: 1
Compactness and guessing principles in the Radin extensions Radin扩展中的紧凑性和猜测原则
IF 0.9 1区 数学
Journal of Mathematical Logic Pub Date : 2021-05-03 DOI: 10.1142/s0219061322500246
Omer Ben-Neria, Jing Zhang
{"title":"Compactness and guessing principles in the Radin extensions","authors":"Omer Ben-Neria, Jing Zhang","doi":"10.1142/s0219061322500246","DOIUrl":"https://doi.org/10.1142/s0219061322500246","url":null,"abstract":"We investigate the interaction between compactness principles and guessing principles in the Radin forcing extensions. In particular, we show that in any Radin forcing extension with respect to a measure sequence on $kappa$, if $kappa$ is weakly compact, then $diamondsuit(kappa)$ holds. This provides contrast with a well-known theorem of Woodin, who showed that in a certain Radin extension over a suitably prepared ground model relative to the existence of large cardinals, the diamond principle fails at a strongly inaccessible Mahlo cardinal. Refining the analysis of the Radin extensions, we consistently demonstrate a scenario where a compactness principle, stronger than the diagonal stationary reflection principle, holds yet the diamond principle fails at a strongly inaccessible cardinal, improving a result from cite{BN19}.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":"26 1","pages":"2250024:1-2250024:22"},"PeriodicalIF":0.9,"publicationDate":"2021-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86425143","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}
引用次数: 4
A negative solution of Kuznetsov's problem for varieties of bi-Heyting algebras 一类双heyting代数的Kuznetsov问题的负解
IF 0.9 1区 数学
Journal of Mathematical Logic Pub Date : 2021-04-13 DOI: 10.1142/s0219061322500131
G. Bezhanishvili, D. Gabelaia, M. Jibladze
{"title":"A negative solution of Kuznetsov's problem for varieties of bi-Heyting algebras","authors":"G. Bezhanishvili, D. Gabelaia, M. Jibladze","doi":"10.1142/s0219061322500131","DOIUrl":"https://doi.org/10.1142/s0219061322500131","url":null,"abstract":"In this paper, we show that there exist (continuum many) varieties of bi-Heyting algebras that are not generated by their complete members. It follows that there exist (continuum many) extensions of the Heyting–Brouwer logic [Formula: see text] that are topologically incomplete. This result provides further insight into the long-standing open problem of Kuznetsov by yielding a negative solution of the reformulation of the problem from extensions of [Formula: see text] to extensions of [Formula: see text].","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":"46 1","pages":"2250013:1-2250013:21"},"PeriodicalIF":0.9,"publicationDate":"2021-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86865007","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}
引用次数: 2
Metrically homogeneous graphs of diameter 3 直径为3的度量齐次图
IF 0.9 1区 数学
Journal of Mathematical Logic Pub Date : 2021-04-01 DOI: 10.1142/s0219061320500208
Daniela A. Amato, G. Cherlin, D. Macpherson
{"title":"Metrically homogeneous graphs of diameter 3","authors":"Daniela A. Amato, G. Cherlin, D. Macpherson","doi":"10.1142/s0219061320500208","DOIUrl":"https://doi.org/10.1142/s0219061320500208","url":null,"abstract":"We classify countable metrically homogeneous graphs of diameter 3.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":"70 1","pages":"2050020:1-2050020:106"},"PeriodicalIF":0.9,"publicationDate":"2021-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83785175","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}
引用次数: 5
The large cardinal strength of weak Vopenka's principle 弱伏本卡原理的大基数强度
IF 0.9 1区 数学
Journal of Mathematical Logic Pub Date : 2021-03-13 DOI: 10.1142/S0219061321500240
Trevor M. Wilson
{"title":"The large cardinal strength of weak Vopenka's principle","authors":"Trevor M. Wilson","doi":"10.1142/S0219061321500240","DOIUrl":"https://doi.org/10.1142/S0219061321500240","url":null,"abstract":"We show that Weak Vopěnka’s Principle, which is the statement that the opposite category of ordinals cannot be fully embedded into the category of graphs, is equivalent to the large cardinal principle Ord is Woodin, which says that for every class [Formula: see text] there is a [Formula: see text]-strong cardinal. Weak Vopěnka’s Principle was already known to imply the existence of a proper class of measurable cardinals. We improve this lower bound to the optimal one by defining structures whose nontrivial homomorphisms can be used as extenders, thereby producing elementary embeddings witnessing [Formula: see text]-strongness of some cardinal.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":"16 1","pages":"2150024:1-2150024:15"},"PeriodicalIF":0.9,"publicationDate":"2021-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80961941","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}
引用次数: 1
Higher indescribability and derived topologies 更高的不可描述性和派生拓扑
IF 0.9 1区 数学
Journal of Mathematical Logic Pub Date : 2021-02-18 DOI: 10.1142/s0219061323500010
Brent Cody
{"title":"Higher indescribability and derived topologies","authors":"Brent Cody","doi":"10.1142/s0219061323500010","DOIUrl":"https://doi.org/10.1142/s0219061323500010","url":null,"abstract":"We introduce reflection properties of cardinals in which the attributes that reflect are expressible by infinitary formulas whose lengths can be strictly larger than the cardinal under consideration. This kind of generalized reflection principle leads to the definitions of $L_{kappa^+,kappa^+}$-indescribability and $Pi^1_xi$-indescribability of a cardinal $kappa$ for all $xi<kappa^+$. In this context, universal $Pi^1_xi$ formulas exist, there is a normal ideal associated to $Pi^1_xi$-indescribability and the notions of $Pi^1_xi$-indescribability yield a strict hierarchy below a measurable cardinal. Additionally, given a regular cardinal $mu$, we introduce a diagonal version of Cantor's derivative operator and use it to extend Bagaria's cite{MR3894041} sequence $langletau_xi:xi<murangle$ of derived topologies on $mu$ to $langletau_xi:xi<mu^+rangle$. Finally, we prove that for all $xi<mu^+$, if there is a stationary set of $alpha<mu$ that have a high enough degree of indescribability, then there are stationarily-many $alpha<mu$ that are nonisolated points in the space $(mu,tau_{xi+1})$.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46870952","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}
引用次数: 0
Simultaneously vanishing higher derived limits without large cardinals 同时消失没有大基数的更高派生极限
IF 0.9 1区 数学
Journal of Mathematical Logic Pub Date : 2021-02-12 DOI: 10.1142/s0219061322500192
J. Bergfalk, M. Hrusák, C. Lambie-Hanson
{"title":"Simultaneously vanishing higher derived limits without large cardinals","authors":"J. Bergfalk, M. Hrusák, C. Lambie-Hanson","doi":"10.1142/s0219061322500192","DOIUrl":"https://doi.org/10.1142/s0219061322500192","url":null,"abstract":"A question dating to Sibe Mardev{s}i'{c} and Andrei Prasolov's 1988 work Strong homology is not additive, and motivating a considerable amount of set theoretic work in the ensuing years, is that of whether it is consistent with the ZFC axioms for the higher derived limits $mathrm{lim}^n$ $(n>0)$ of a certain inverse system $mathbf{A}$ indexed by ${^omega}omega$ to simultaneously vanish. An equivalent formulation of this question is that of whether it is consistent for all $n$-coherent families of functions indexed by ${^omega}omega$ to be trivial. In this paper, we prove that, in any forcing extension given by adjoining $beth_omega$-many Cohen reals, $mathrm{lim}^n mathbf{A}$ vanishes for all $n>0$. Our proof involves a detailed combinatorial analysis of the forcing extension and repeated applications of higher dimensional $Delta$-system lemmas. This work removes all large cardinal hypotheses from the main result of arXiv:1907.11744 and substantially reduces the least value of the continuum known to be compatible with the simultaneous vanishing of $mathrm{lim}^n mathbf{A}$ for all $n>0$.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":"16 1","pages":"2250019:1-2250019:40"},"PeriodicalIF":0.9,"publicationDate":"2021-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82652032","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}
引用次数: 7
Incompatible bounded category forcing axioms 不相容有界范畴强迫公理
IF 0.9 1区 数学
Journal of Mathematical Logic Pub Date : 2021-01-08 DOI: 10.1142/s0219061322500064
D. Asperó, M. Viale
{"title":"Incompatible bounded category forcing axioms","authors":"D. Asperó, M. Viale","doi":"10.1142/s0219061322500064","DOIUrl":"https://doi.org/10.1142/s0219061322500064","url":null,"abstract":"We introduce bounded category forcing axioms for well-behaved classes [Formula: see text]. These are strong forms of bounded forcing axioms which completely decide the theory of some initial segment of the universe [Formula: see text] modulo forcing in [Formula: see text], for some cardinal [Formula: see text] naturally associated to [Formula: see text]. These axioms naturally extend projective absoluteness for arbitrary set-forcing — in this situation [Formula: see text] — to classes [Formula: see text] with [Formula: see text]. Unlike projective absoluteness, these higher bounded category forcing axioms do not follow from large cardinal axioms but can be forced under mild large cardinal assumptions on [Formula: see text]. We also show the existence of many classes [Formula: see text] with [Formula: see text] giving rise to pairwise incompatible theories for [Formula: see text].","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":"8 1","pages":"2250006:1-2250006:76"},"PeriodicalIF":0.9,"publicationDate":"2021-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86208738","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}
引用次数: 4
Erratum: Equations in oligomorphic clones and the constraint satisfaction problem for ω-categorical structures 勘误:低纯克隆方程和ω-范畴结构的约束满足问题
IF 0.9 1区 数学
Journal of Mathematical Logic Pub Date : 2021-01-01 DOI: 10.1142/s021906132192001x
L. Barto, M. Kompatscher, M. Olsák, Trung Van Pham, M. Pinsker
{"title":"Erratum: Equations in oligomorphic clones and the constraint satisfaction problem for ω-categorical structures","authors":"L. Barto, M. Kompatscher, M. Olsák, Trung Van Pham, M. Pinsker","doi":"10.1142/s021906132192001x","DOIUrl":"https://doi.org/10.1142/s021906132192001x","url":null,"abstract":"","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":"51 1","pages":"2192001:1"},"PeriodicalIF":0.9,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90356660","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}
引用次数: 0
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