In this paper we investigate Schur ultrafilters on groups. Using the algebraic structure of Stone-v{C}ech compactifications of discrete groups and Schur ultrafilters, we give a new description of Bohr compactifications of topological groups. This approach allows us to characterize chart groups that are topological groups. Namely, a chart group $G$ is a topological group if and only if each Schur ultrafilter on $G$ converges to the unit of $G$.
{"title":"Schur ultrafilters and Bohr compactifications of topological groups","authors":"Serhii Bardyla, Pavol Zlatoš","doi":"arxiv-2409.07280","DOIUrl":"https://doi.org/arxiv-2409.07280","url":null,"abstract":"In this paper we investigate Schur ultrafilters on groups. Using the\u0000algebraic structure of Stone-v{C}ech compactifications of discrete groups and\u0000Schur ultrafilters, we give a new description of Bohr compactifications of\u0000topological groups. This approach allows us to characterize chart groups that\u0000are topological groups. Namely, a chart group $G$ is a topological group if and\u0000only if each Schur ultrafilter on $G$ converges to the unit of $G$.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":"32 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187706","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}
Kinna--Wagner Principles state that every set can be mapped into some fixed iterated power set of an ordinal, and we write $mathsf{KWP}$ to denote that there is some $alpha$ for which this holds. The Kinna--Wagner Conjecture, formulated by the first author in [9], states that if $V$ is a model of $mathsf{ZF+KWP}$ and $G$ is a $V$-generic filter, then whenever $W$ is an intermediate model of $mathsf{ZF}$, that is $Vsubseteq Wsubseteq V[G]$, then $W=V(x)$ for some $x$ if and only if $W$ satisfies $mathsf{KWP}$. In this work we prove the conjecture and generalise it even further. We include a brief historical overview of Kinna--Wagner Principles and new results about Kinna--Wagner Principles in the multiverse of sets.
{"title":"Intermediate models and Kinna--Wagner Principles","authors":"Asaf Karagila, Jonathan Schilhan","doi":"arxiv-2409.07352","DOIUrl":"https://doi.org/arxiv-2409.07352","url":null,"abstract":"Kinna--Wagner Principles state that every set can be mapped into some fixed\u0000iterated power set of an ordinal, and we write $mathsf{KWP}$ to denote that\u0000there is some $alpha$ for which this holds. The Kinna--Wagner Conjecture,\u0000formulated by the first author in [9], states that if $V$ is a model of\u0000$mathsf{ZF+KWP}$ and $G$ is a $V$-generic filter, then whenever $W$ is an\u0000intermediate model of $mathsf{ZF}$, that is $Vsubseteq Wsubseteq V[G]$, then\u0000$W=V(x)$ for some $x$ if and only if $W$ satisfies $mathsf{KWP}$. In this work\u0000we prove the conjecture and generalise it even further. We include a brief\u0000historical overview of Kinna--Wagner Principles and new results about\u0000Kinna--Wagner Principles in the multiverse of sets.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":"52 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142224551","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}
We present an $L$-like construction that produces the minimal model of $mathsf{AD}_mathbb{R}+$"$Theta$ is regular". In fact, our construction can produce any model of $mathsf{AD}^++mathsf{AD}_mathbb{R}+V=L(mathcal{P}(mathbb{R}))$ in which there is no hod mouse with a measurable limit of Woodins.
{"title":"Building Models of Determinacy from Below","authors":"Obrad Kasum, Grigor Sargsyan","doi":"arxiv-2409.07156","DOIUrl":"https://doi.org/arxiv-2409.07156","url":null,"abstract":"We present an $L$-like construction that produces the minimal model of\u0000$mathsf{AD}_mathbb{R}+$\"$Theta$ is regular\". In fact, our construction can\u0000produce any model of\u0000$mathsf{AD}^++mathsf{AD}_mathbb{R}+V=L(mathcal{P}(mathbb{R}))$ in which\u0000there is no hod mouse with a measurable limit of Woodins.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187705","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}
Manuel Bodirsky, Édouard Bonnet, Žaneta Semanišinová
We study the complexity of the valued constraint satisfaction problem (VCSP) for every valued structure with the domain ${mathbb Q}$ that is preserved by all order-preserving bijections. Such VCSPs will be called temporal, in analogy to the (classical) constraint satisfaction problem: a relational structure is preserved by all order-preserving bijections if and only if all its relations have a first-order definition in $({mathbb Q};<)$, and the CSPs for such structures are called temporal CSPs. Many optimization problems that have been studied intensively in the literature can be phrased as a temporal VCSP. We prove that a temporal VCSP is in P, or NP-complete. Our analysis uses the concept of fractional polymorphisms; this is the first dichotomy result for VCSPs over infinite domains which is complete in the sense that it treats all valued structures with a given automorphism group.
{"title":"Temporal Valued Constraint Satisfaction Problems","authors":"Manuel Bodirsky, Édouard Bonnet, Žaneta Semanišinová","doi":"arxiv-2409.07285","DOIUrl":"https://doi.org/arxiv-2409.07285","url":null,"abstract":"We study the complexity of the valued constraint satisfaction problem (VCSP)\u0000for every valued structure with the domain ${mathbb Q}$ that is preserved by\u0000all order-preserving bijections. Such VCSPs will be called temporal, in analogy\u0000to the (classical) constraint satisfaction problem: a relational structure is\u0000preserved by all order-preserving bijections if and only if all its relations\u0000have a first-order definition in $({mathbb Q};<)$, and the CSPs for such\u0000structures are called temporal CSPs. Many optimization problems that have been\u0000studied intensively in the literature can be phrased as a temporal VCSP. We\u0000prove that a temporal VCSP is in P, or NP-complete. Our analysis uses the\u0000concept of fractional polymorphisms; this is the first dichotomy result for\u0000VCSPs over infinite domains which is complete in the sense that it treats all\u0000valued structures with a given automorphism group.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":"5 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187704","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}
We introduce and study a notion of Borel order dimension for Borel quasi orders. It will be shown that this notion is closely related to the notion of Borel dichromatic number for simple directed graphs. We prove a dichotomy, which generalizes the ${GGG}_{0}$-dichotomy, for the Borel dichromatic number of Borel simple directed graphs. By applying this dichotomy to Borel quasi orders, another dichotomy that characterizes the Borel quasi orders of uncountable Borel dimension is proved. We obtain further structural information about the Borel quasi orders of countable Borel dimension by showing that they are all Borel linearizable. We then investigate the locally countable Borel quasi orders in more detail, paying special attention to the Turing degrees, and produce models of set theory where the continuum is arbitrarily large and all locally countable Borel quasi orders are of Borel dimension less than the continuum. Combining our results here with earlier work shows that the Borel order dimension of the Turing degrees is usually strictly larger than its classical order dimension.
{"title":"Borel Order Dimension","authors":"Dilip Raghavan, Ming Xiao","doi":"arxiv-2409.06516","DOIUrl":"https://doi.org/arxiv-2409.06516","url":null,"abstract":"We introduce and study a notion of Borel order dimension for Borel quasi\u0000orders. It will be shown that this notion is closely related to the notion of\u0000Borel dichromatic number for simple directed graphs. We prove a dichotomy,\u0000which generalizes the ${GGG}_{0}$-dichotomy, for the Borel dichromatic number\u0000of Borel simple directed graphs. By applying this dichotomy to Borel quasi\u0000orders, another dichotomy that characterizes the Borel quasi orders of\u0000uncountable Borel dimension is proved. We obtain further structural information\u0000about the Borel quasi orders of countable Borel dimension by showing that they\u0000are all Borel linearizable. We then investigate the locally countable Borel\u0000quasi orders in more detail, paying special attention to the Turing degrees,\u0000and produce models of set theory where the continuum is arbitrarily large and\u0000all locally countable Borel quasi orders are of Borel dimension less than the\u0000continuum. Combining our results here with earlier work shows that the Borel\u0000order dimension of the Turing degrees is usually strictly larger than its\u0000classical order dimension.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187707","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}
Samson Alva, Eduardo Dueñez, Jose Iovino, Claire Walton
We introduce a general concept of layered computation model, of which neural networks are a particular example, and combine tools of topological dynamics and model theory to study asymptotics of such models. We prove that, as the number of layers of a computation grows, the computation reaches a state of ``deep equilibrium" which amounts to a single, self-referential layer. After proving the existence of deep equilibria under fairly general hypotheses, we characterize their computability.
{"title":"Deep Equilibria: Existence and Computability","authors":"Samson Alva, Eduardo Dueñez, Jose Iovino, Claire Walton","doi":"arxiv-2409.06064","DOIUrl":"https://doi.org/arxiv-2409.06064","url":null,"abstract":"We introduce a general concept of layered computation model, of which neural\u0000networks are a particular example, and combine tools of topological dynamics\u0000and model theory to study asymptotics of such models. We prove that, as the\u0000number of layers of a computation grows, the computation reaches a state of\u0000``deep equilibrium\" which amounts to a single, self-referential layer. After\u0000proving the existence of deep equilibria under fairly general hypotheses, we\u0000characterize their computability.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":"15 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187710","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}
For which choices of $X,Y,Zin{Sigma^1_1,Pi^1_1}$ does no sufficiently strong $X$-sound and $Y$-definable extension theory prove its own $Z$-soundness? We give a complete answer, thereby delimiting the generalizations of G"odel's second incompleteness theorem that hold within second-order arithmetic.
{"title":"A classification of incompleteness statements","authors":"Henry Towsner, James Walsh","doi":"arxiv-2409.05973","DOIUrl":"https://doi.org/arxiv-2409.05973","url":null,"abstract":"For which choices of $X,Y,Zin{Sigma^1_1,Pi^1_1}$ does no sufficiently\u0000strong $X$-sound and $Y$-definable extension theory prove its own\u0000$Z$-soundness? We give a complete answer, thereby delimiting the\u0000generalizations of G\"odel's second incompleteness theorem that hold within\u0000second-order arithmetic.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":"59 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187708","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}
Esakia's theorem states that Grzegorczyk's logic is the largest modal companion of intuitionistic propositional calculus. We prove that already the one-variable fragment of intuitionistic predicate calculus does not have the largest modal companion, yielding that Esakia's theorem fails in the monadic setting.
{"title":"Failure of Esakia's theorem in the monadic setting","authors":"Guram Bezhanishvili, Luca Carai","doi":"arxiv-2409.05607","DOIUrl":"https://doi.org/arxiv-2409.05607","url":null,"abstract":"Esakia's theorem states that Grzegorczyk's logic is the largest modal\u0000companion of intuitionistic propositional calculus. We prove that already the\u0000one-variable fragment of intuitionistic predicate calculus does not have the\u0000largest modal companion, yielding that Esakia's theorem fails in the monadic\u0000setting.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":"5 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187711","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}
We prove various results around indiscernibles in monadically NIP theories. First, we provide several characterizations of monadic NIP in terms of indiscernibles, mirroring previous characterizations in terms of the behavior of finite satisfiability. Second, we study (monadic) distality in hereditary classes and complete theories. Here, via finite combinatorics, we prove a result implying that every planar graph admits a distal expansion. Finally, we prove a result implying that no monadically NIP theory interprets an infinite group, and note an example of a (monadically) stable theory with no distal expansion that does not interpret an infinite group.
{"title":"Indiscernibles in monadically NIP theories","authors":"Samuel Braunfeld, Michael C. Laskowski","doi":"arxiv-2409.05223","DOIUrl":"https://doi.org/arxiv-2409.05223","url":null,"abstract":"We prove various results around indiscernibles in monadically NIP theories.\u0000First, we provide several characterizations of monadic NIP in terms of\u0000indiscernibles, mirroring previous characterizations in terms of the behavior\u0000of finite satisfiability. Second, we study (monadic) distality in hereditary\u0000classes and complete theories. Here, via finite combinatorics, we prove a\u0000result implying that every planar graph admits a distal expansion. Finally, we\u0000prove a result implying that no monadically NIP theory interprets an infinite\u0000group, and note an example of a (monadically) stable theory with no distal\u0000expansion that does not interpret an infinite group.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":"18 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187731","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}
T-convergence groups is a natural extension of lattice-valued topological groups, which is a newly introduced mathematical structure. In this paper, we will further explore the theory of T-convergence groups. The main results include: (1) It possesses a novel characterization through the $odot$-product of T-filters, and it is localizable, meaning that each T-convergence group is uniquely determined by the convergence at the identity element of the underlying group. (2) The definition of its subcategory, the topological T-convergence groups, can be simplified by removing the topological condition (TT). (3) It exhibits uniformization, which means that each T-convergence group can be reconstructed from a T-uniformly convergent space. (4) It possesses a power object, indicating that it has good category properties.
T-收敛群是格值拓扑群的自然扩展，是一种新引入的数学结构。本文将进一步探讨 T 趋近群的理论。主要结果包括(1）通过 T 滤波的 $odot$ 产物，它拥有一个新颖的表征，并且它是可局部化的，这意味着每个 T 收敛群都是由底层群的标识元处的收敛所唯一决定的。(2）它的子类拓扑 T- 收敛群的定义可以通过去掉拓扑条件（TT）来简化。(3) 它具有均匀性，即每个 T 收敛群都可以从一个 T 均匀收敛空间重建。(4）它具有幂对象，表明它具有良好的范畴性质。
{"title":"The further study on the category of T-convergence groups","authors":"Lingqiang Li, Qiu Jin","doi":"arxiv-2409.04939","DOIUrl":"https://doi.org/arxiv-2409.04939","url":null,"abstract":"T-convergence groups is a natural extension of lattice-valued topological\u0000groups, which is a newly introduced mathematical structure. In this paper, we\u0000will further explore the theory of T-convergence groups. The main results\u0000include: (1) It possesses a novel characterization through the $odot$-product\u0000of T-filters, and it is localizable, meaning that each T-convergence group is\u0000uniquely determined by the convergence at the identity element of the\u0000underlying group. (2) The definition of its subcategory, the topological\u0000T-convergence groups, can be simplified by removing the topological condition\u0000(TT). (3) It exhibits uniformization, which means that each T-convergence group\u0000can be reconstructed from a T-uniformly convergent space. (4) It possesses a\u0000power object, indicating that it has good category properties.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":"184 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142224552","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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