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Journal of Mathematical Logic最新文献

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Borel combinatorics fail in HYP Borel组合在HYP中失败
IF 0.9 1区 数学
Journal of Mathematical Logic Pub Date : 2021-06-24 DOI: 10.1142/s0219061322500234
H. Towsner, Rose Weisshaar, L. Westrick
{"title":"Borel combinatorics fail in HYP","authors":"H. Towsner, Rose Weisshaar, L. Westrick","doi":"10.1142/s0219061322500234","DOIUrl":"https://doi.org/10.1142/s0219061322500234","url":null,"abstract":"We characterize the completely determined Borel subsets of HYP as exactly the omega_1^{ck} subsets of HYP. As a result, HYP believes there is a Borel well-ordering of the reals, that the Borel Dual Ramsey Theorem fails, and that every Borel d-regular bipartite graph has a Borel perfect matching, among other examples. Therefore, the Borel Dual Ramsey Theorem and several theorems of descriptive combinatorics are not theories of hyperarithmetic analysis. In the case of the Borel Dual Ramsey Theorem, this answers a question of Astor, Dzhafarov, Montalban, Solomon&the third author.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2021-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80975099","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
On the antichain tree property 关于反链树的性质
IF 0.9 1区 数学
Journal of Mathematical Logic Pub Date : 2021-06-07 DOI: 10.1142/s0219061322500210
Jinhoo Ahn, J. Kim, Junguk Lee
{"title":"On the antichain tree property","authors":"Jinhoo Ahn, J. Kim, Junguk Lee","doi":"10.1142/s0219061322500210","DOIUrl":"https://doi.org/10.1142/s0219061322500210","url":null,"abstract":"In this note, we investigate a new model theoretical tree property, called the antichain tree property (ATP). We develop combinatorial techniques for ATP. First, we show that ATP is always witnessed by a formula in a single free variable, and for formulas, not having ATP is closed under disjunction. Second, we show the equivalence of ATP and $k$-ATP, and provide a criterion for theories to have not ATP (being NATP). Using these combinatorial observations, we find algebraic examples of ATP and NATP, including pure group, pure fields, and valued fields. More precisely, we prove Mekler's construction for groups, Chatzidakis' style criterion for PAC fields, and the AKE-style principle for valued fields preserving NATP. And we give a construction of an antichain tree in the Skolem arithmetic and atomless Boolean algebras.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2021-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79924985","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: 12
A generalization of the 픾0 dichotomy and a strengthening of the 피0ℕ dichotomy 对픾0二分法的推广和对_(_)0二分法的强化
IF 0.9 1区 数学
Journal of Mathematical Logic Pub Date : 2021-05-31 DOI: 10.1142/S0219061321500288
B. D. Miller
{"title":"A generalization of the 픾0 dichotomy and a strengthening of the 피0ℕ dichotomy","authors":"B. D. Miller","doi":"10.1142/S0219061321500288","DOIUrl":"https://doi.org/10.1142/S0219061321500288","url":null,"abstract":"We generalize the [Formula: see text] dichotomy to doubly-indexed sequences of analytic digraphs. Under a mild definability assumption, we use this generalization to characterize the family of Borel actions of tsi Polish groups on Polish spaces that can be decomposed into countably-many Borel actions admitting complete Borel sets that are lacunary with respect to an open neighborhood of the identity. We also show that if the group in question is non-archimedean, then the inexistence of such a decomposition yields a special kind of continuous embedding of [Formula: see text] into the corresponding orbit equivalence relation.","PeriodicalId":50144,"journal":{"name":"Journal of Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2021-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80366992","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
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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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
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