Pub Date : 2021-05-15DOI: 10.1142/s0219061322500258
A. Chernikov, E. Hrushovski, Alex Kruckman, K. Krupiński, Slavko Moconja, A. Pillay, N. Ramsey
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.
{"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}
Pub Date : 2021-05-11DOI: 10.1142/S0219061322500283
Shaun Allison, Assaf Shani
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.
{"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}
Pub Date : 2021-05-03DOI: 10.1142/s0219061322500246
Omer Ben-Neria, Jing Zhang
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}.
{"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}
Pub Date : 2021-04-13DOI: 10.1142/s0219061322500131
G. Bezhanishvili, D. Gabelaia, M. Jibladze
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].
{"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}
Pub Date : 2021-04-01DOI: 10.1142/s0219061320500208
Daniela A. Amato, G. Cherlin, D. Macpherson
We classify countable metrically homogeneous graphs of diameter 3.
我们对直径为3的可数度量齐次图进行分类。
{"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}
Pub Date : 2021-03-13DOI: 10.1142/S0219061321500240
Trevor M. Wilson
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.
我们证明弱vopovinka原理，即序数的相反类别不能完全嵌入图的类别，相当于大基数原理Ord is Woodin，它说对于每个类[公式:见文]都有一个[公式:见文]-强基数。人们已经知道，弱沃普涅卡原理暗示存在一类适当的可测基数。我们通过定义非平凡同态可以用作扩展器的结构，将这个下界改进为最优下界，从而产生初等嵌入，证明[公式:见文本]-一些基数的强度。
{"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}
Pub Date : 2021-02-18DOI: 10.1142/s0219061323500010
Brent Cody
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
{"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}
Pub Date : 2021-02-12DOI: 10.1142/s0219061322500192
J. Bergfalk, M. Hrusák, C. Lambie-Hanson
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 $n>0$.
Sibe Marde v{s}和Andrei Prasolov在1988年的著作《强同调不是可加的》(Strong homology is not additive)中提出的一个问题是，对于以${^omega}omega$为索引的某逆系统$mathbf{A}$的较高推导极限$mathrm{lim}^n$$(n>0)$是否与ZFC公理相一致，这个问题在随后的几年里激发了大量的集合论工作。这个问题的一个等价的表述是，是否所有$n$ -相干族的函数都以${^omega}omega$为索引是平凡的。在本文中，我们证明了在任意由相邻的$beth_omega$ -多个Cohen实数给出的强迫扩展中，对于所有$n>0$, $mathrm{lim}^n mathbf{A}$都消失。我们的证明包括对高维$Delta$ -系统引理的强迫扩展和重复应用的详细组合分析。这项工作从arXiv:1907.11744的主要结果中删除了所有大的基本假设，并大大降低了已知与所有$n>0$的$mathrm{lim}^n mathbf{A}$同时消失相容的连续统的最小值。
{"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}
Pub Date : 2021-01-08DOI: 10.1142/s0219061322500064
D. Asperó, M. Viale
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].
{"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}
Pub Date : 2021-01-01DOI: 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}
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