An infinite game on the set of real numbers appeared in Matthew Baker's work [Math. Mag. 80 (2007), no. 5, pp. 377--380] in which he asks whether it can help characterize countable subsets of the reals. This question is in a similar spirit to how the Banach-Mazur Game characterizes meager sets in an arbitrary topological space. In a recent paper, Will Brian and Steven Clontz prove that in Baker's game, Player II has a winning strategy if and only if the payoff set is countable. They also asked if it is possible, in general linear orders, for Player II to have a winning strategy on some uncountable set. To this we give a positive answer and moreover construct, for every infinite cardinal $kappa$, a dense linear order of size $kappa$ on which Player II has a winning strategy on all payoff sets. We finish with some future research questions, further underlining the difficulty in generalizing the characterization of Brian and Clontz to linear orders.
{"title":"Uncountable sets and an infinite linear order game","authors":"Tonatiuh Matos-Wiederhold, Luciano Salvetti","doi":"arxiv-2408.14624","DOIUrl":"https://doi.org/arxiv-2408.14624","url":null,"abstract":"An infinite game on the set of real numbers appeared in Matthew Baker's work\u0000[Math. Mag. 80 (2007), no. 5, pp. 377--380] in which he asks whether it can\u0000help characterize countable subsets of the reals. This question is in a similar\u0000spirit to how the Banach-Mazur Game characterizes meager sets in an arbitrary\u0000topological space. In a recent paper, Will Brian and Steven Clontz prove that in Baker's game,\u0000Player II has a winning strategy if and only if the payoff set is countable.\u0000They also asked if it is possible, in general linear orders, for Player II to\u0000have a winning strategy on some uncountable set. To this we give a positive answer and moreover construct, for every infinite\u0000cardinal $kappa$, a dense linear order of size $kappa$ on which Player II has\u0000a winning strategy on all payoff sets. We finish with some future research\u0000questions, further underlining the difficulty in generalizing the\u0000characterization of Brian and Clontz to linear orders.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187775","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}
Cut-elimination theorems constitute one of the most important classes of theorems of proof theory. Since Gentzen's proof of the cut-elimination theorem for the system $mathbf{LK}$, several other proofs have been proposed. Even though the techniques of these proofs can be modified to sequent systems other than $mathbf{LK}$, they are essentially of a very particular nature; each of them describes an algorithm to transform a given proof to a cut-free proof. However, due to its reliance on heavy syntactic arguments and case distinctions, such an algorithm makes the fundamental structure of the argument rather opaque. We, therefore, consider rules abstractly, within the framework of logical structures familiar from universal logic `a la Jean-Yves B'eziau, and aim to clarify the essence of the so-called ``elimination theorems''. To do this, we first give a non-algorithmic proof of the cut-elimination theorem for the propositional fragment of $mathbf{LK}$. From this proof, we abstract the essential features of the argument and define something called ``normal sequent structures'' relative to a particular rule. We then prove a version of the rule-elimination theorem for these. Abstracting even more, we define ``abstract sequent structures'' and show that for these structures, the corresponding version of the ``rule''-elimination theorem has a converse as well.
{"title":"Rule-Elimination Theorems","authors":"Sayantan Roy","doi":"arxiv-2408.14581","DOIUrl":"https://doi.org/arxiv-2408.14581","url":null,"abstract":"Cut-elimination theorems constitute one of the most important classes of\u0000theorems of proof theory. Since Gentzen's proof of the cut-elimination theorem\u0000for the system $mathbf{LK}$, several other proofs have been proposed. Even\u0000though the techniques of these proofs can be modified to sequent systems other\u0000than $mathbf{LK}$, they are essentially of a very particular nature; each of\u0000them describes an algorithm to transform a given proof to a cut-free proof.\u0000However, due to its reliance on heavy syntactic arguments and case\u0000distinctions, such an algorithm makes the fundamental structure of the argument\u0000rather opaque. We, therefore, consider rules abstractly, within the framework\u0000of logical structures familiar from universal logic `a la Jean-Yves B'eziau,\u0000and aim to clarify the essence of the so-called ``elimination theorems''. To do\u0000this, we first give a non-algorithmic proof of the cut-elimination theorem for\u0000the propositional fragment of $mathbf{LK}$. From this proof, we abstract the\u0000essential features of the argument and define something called ``normal sequent\u0000structures'' relative to a particular rule. We then prove a version of the\u0000rule-elimination theorem for these. Abstracting even more, we define ``abstract\u0000sequent structures'' and show that for these structures, the corresponding\u0000version of the ``rule''-elimination theorem has a converse as well.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187756","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}
Consider a definable complete d-minimal expansion $(F, <, +, cdot, 0, 1, dots,)$ of an oredered field $F$. Let $X$ be a definably compact definably normal definable $C^r$ manifold and $2 le r
考虑一个有序域 $F$ 的可定义完整 d 最小展开 $(F, <, +, cdot, 0, 1,dots,)$ 。让 $X$ 是一个可定义的紧凑可定义的正常可定义的 $C^r$ 流形，且 $2 le r
{"title":"Morse theory in definably complete d-minimal structures","authors":"Masato Fujita, Tomohiro Kawakami","doi":"arxiv-2408.14675","DOIUrl":"https://doi.org/arxiv-2408.14675","url":null,"abstract":"Consider a definable complete d-minimal expansion $(F, <, +, cdot, 0, 1,\u0000dots,)$ of an oredered field $F$. Let $X$ be a definably compact definably\u0000normal definable $C^r$ manifold and $2 le r <infty$. We prove that the set of\u0000definable Morse functions is open and dense in the set of definable $C^r$\u0000functions on $X$ with respect to the definable $C^2$ topology.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187755","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}
Sayantan Roy, Sankha S. Basu, Mihir K. Chakraborty
In this article, we try to formulate a definition of ''many-valued logical structure''. For this, we embark on a deeper study of Suszko's Thesis ($mathbf{ST}$) and show that the truth or falsity of $mathbf{ST}$ depends, at least, on the precise notion of semantics. We propose two different notions of semantics and three different notions of entailment. The first one helps us formulate a precise definition of inferentially many-valued logical structures. The second and the third help us to generalise Suszko Reduction and provide adequate bivalent semantics for monotonic and a couple of nonmonotonic logical structures. All these lead us to a closer examination of the played by language/metalanguage hierarchy vis-'a-vis $mathbf{ST}$. We conclude that many-valued logical structures can be obtained if the bivalence of all the higher-order metalogics of the logic under consideration is discarded, building formal bridges between the theory of graded consequence and the theory of many-valued logical structures, culminating in generalisations of Suszko's Thesis.
{"title":"Suszko's Thesis and Many-valued Logical Structures","authors":"Sayantan Roy, Sankha S. Basu, Mihir K. Chakraborty","doi":"arxiv-2408.13769","DOIUrl":"https://doi.org/arxiv-2408.13769","url":null,"abstract":"In this article, we try to formulate a definition of ''many-valued logical\u0000structure''. For this, we embark on a deeper study of Suszko's Thesis\u0000($mathbf{ST}$) and show that the truth or falsity of $mathbf{ST}$ depends, at\u0000least, on the precise notion of semantics. We propose two different notions of\u0000semantics and three different notions of entailment. The first one helps us\u0000formulate a precise definition of inferentially many-valued logical structures.\u0000The second and the third help us to generalise Suszko Reduction and provide\u0000adequate bivalent semantics for monotonic and a couple of nonmonotonic logical\u0000structures. All these lead us to a closer examination of the played by\u0000language/metalanguage hierarchy vis-'a-vis $mathbf{ST}$. We conclude that\u0000many-valued logical structures can be obtained if the bivalence of all the\u0000higher-order metalogics of the logic under consideration is discarded, building\u0000formal bridges between the theory of graded consequence and the theory of\u0000many-valued logical structures, culminating in generalisations of Suszko's\u0000Thesis.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187777","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}
Going back to Kreisel in the Sixties, hyperarithmetical analysis is a cluster of logical systems just beyond arithmetical comprehension. Only recently natural examples of theorems from the mathematical mainstream were identified that fit this category. In this paper, we provide many examples of theorems of real analysis that sit within the range of hyperarithmetical analysis, namely between the higher-order version of $Sigma_1^1$-AC$_0$ and weak-$Sigma_1^1$-AC$_0$, working in Kohlenbach's higher-order framework. Our example theorems are based on the Jordan decomposition theorem, unordered sums, metric spaces, and semi-continuous functions. Along the way, we identify a couple of new systems of hyperarithmetical analysis.
{"title":"Connecting real and hyperarithmetical analysis","authors":"Sam Sanders","doi":"arxiv-2408.13760","DOIUrl":"https://doi.org/arxiv-2408.13760","url":null,"abstract":"Going back to Kreisel in the Sixties, hyperarithmetical analysis is a cluster\u0000of logical systems just beyond arithmetical comprehension. Only recently\u0000natural examples of theorems from the mathematical mainstream were identified\u0000that fit this category. In this paper, we provide many examples of theorems of\u0000real analysis that sit within the range of hyperarithmetical analysis, namely\u0000between the higher-order version of $Sigma_1^1$-AC$_0$ and\u0000weak-$Sigma_1^1$-AC$_0$, working in Kohlenbach's higher-order framework. Our\u0000example theorems are based on the Jordan decomposition theorem, unordered sums,\u0000metric spaces, and semi-continuous functions. Along the way, we identify a\u0000couple of new systems of hyperarithmetical analysis.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187778","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}
Every Cauchy sequence is definable in a d-minimal expansion of the $mathbb R$-vector space over $mathbb R$. In this paper, we prove this assertion and the following more general assertion: Let $mathcal R$ be either the ordered $mathbb R$-vector space structure over $mathbb R$ or the ordered group of reals. A first-order expansion of $mathcal R$ by a countable subset $D$ of $mathbb R$ and a compact subset $E$ of $mathbb R$ of finite Cantor-Bendixson rank is d-minimal if $(mathcal R,D)$ is locally o-minimal.
{"title":"Every Cauchy sequence is definable in a d-minimal expansion of the $mathbb R$-vector space over $mathbb R$","authors":"Masato Fujita","doi":"arxiv-2408.12883","DOIUrl":"https://doi.org/arxiv-2408.12883","url":null,"abstract":"Every Cauchy sequence is definable in a d-minimal expansion of the $mathbb\u0000R$-vector space over $mathbb R$. In this paper, we prove this assertion and\u0000the following more general assertion: Let $mathcal R$ be either the ordered\u0000$mathbb R$-vector space structure over $mathbb R$ or the ordered group of\u0000reals. A first-order expansion of $mathcal R$ by a countable subset $D$ of\u0000$mathbb R$ and a compact subset $E$ of $mathbb R$ of finite Cantor-Bendixson\u0000rank is d-minimal if $(mathcal R,D)$ is locally o-minimal.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187795","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}
Christoph Berkholz, Dietrich Kuske, Christian Schwarz
Is it possible to write significantly smaller formulae when using Boolean operators other than those of the De Morgan basis (and, or, not, and the constants)? For propositional logic, a negative answer was given by Pratt: formulae over one set of operators can always be translated into an equivalent formula over any other complete set of operators with only polynomial increase in size. Surprisingly, for modal logic the picture is different: we show that elimination of bi-implication is only possible at the cost of an exponential number of occurrences of the modal operator $lozenge$ and therefore of an exponential increase in formula size, i.e., the De Morgan basis and its extension by bi-implication differ in succinctness. Moreover, we prove that any complete set of Boolean operators agrees in succinctness with the De Morgan basis or with its extension by bi-implication. More precisely, these results are shown for the modal logic $mathrm{T}$ (and therefore for $mathrm{K}$). We complement them showing that the modal logic $mathrm{S5}$ behaves as propositional logic: the choice of Boolean operators has no significant impact on the size of formulae.
{"title":"Boolean basis, formula size, and number of modal operators","authors":"Christoph Berkholz, Dietrich Kuske, Christian Schwarz","doi":"arxiv-2408.11651","DOIUrl":"https://doi.org/arxiv-2408.11651","url":null,"abstract":"Is it possible to write significantly smaller formulae when using Boolean\u0000operators other than those of the De Morgan basis (and, or, not, and the\u0000constants)? For propositional logic, a negative answer was given by Pratt:\u0000formulae over one set of operators can always be translated into an equivalent\u0000formula over any other complete set of operators with only polynomial increase\u0000in size. Surprisingly, for modal logic the picture is different: we show that\u0000elimination of bi-implication is only possible at the cost of an exponential\u0000number of occurrences of the modal operator $lozenge$ and therefore of an\u0000exponential increase in formula size, i.e., the De Morgan basis and its\u0000extension by bi-implication differ in succinctness. Moreover, we prove that any\u0000complete set of Boolean operators agrees in succinctness with the De Morgan\u0000basis or with its extension by bi-implication. More precisely, these results\u0000are shown for the modal logic $mathrm{T}$ (and therefore for $mathrm{K}$). We\u0000complement them showing that the modal logic $mathrm{S5}$ behaves as\u0000propositional logic: the choice of Boolean operators has no significant impact\u0000on the size of formulae.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187780","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}
Any simple pseudofinite group G is known to be isomorphic to a (twisted) Chevalley group over a pseudofinite field. This celebrated result mostly follows from the work of Wilson in 1995 and heavily relies on the classification of finite simple groups (CFSG). It easily follows that G is finite-dimensional with additive and fine dimension and, in particular, that if dim(G)=3 then G is isomorphic to PSL(2,F) for some pseudofinite field F. We describe pseudofinite finite-dimensional groups when the dimension is fine, additive and <4 and, in particular, show that the classification G isomorphic to PSL(2,F) is independent from CFSG.
众所周知，任何单假无限群 G 都与假无限域上的（扭曲）切瓦利群同构。这一著名的结果主要源于威尔逊在 1995 年的工作，并在很大程度上依赖于有限简单群（CFSG）的分类。我们描述了当维度为细维度、加维度和 <4 时的伪有限维群，并特别证明了 G 与 PSL(2,F) 的同构分类与 CFSG 无关。
{"title":"Finite-dimensional pseudofinite groups of small dimension, without CFSG","authors":"Ulla KarhumäkiAGL, Frank Olaf WagnerAGL","doi":"arxiv-2408.11484","DOIUrl":"https://doi.org/arxiv-2408.11484","url":null,"abstract":"Any simple pseudofinite group G is known to be isomorphic to a (twisted)\u0000Chevalley group over a pseudofinite field. This celebrated result mostly\u0000follows from the work of Wilson in 1995 and heavily relies on the\u0000classification of finite simple groups (CFSG). It easily follows that G is\u0000finite-dimensional with additive and fine dimension and, in particular, that if\u0000dim(G)=3 then G is isomorphic to PSL(2,F) for some pseudofinite field F. We\u0000describe pseudofinite finite-dimensional groups when the dimension is fine,\u0000additive and <4 and, in particular, show that the classification G isomorphic\u0000to PSL(2,F) is independent from CFSG.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187781","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}
Sub-sub-intuitionistic logic is obtained from intuitionistic logic by weakening the implication and removing distributivity. It can alternatively be viewed as conditional weak positive logic. We provide semantics for sub-sub-intuitionistic logic by means of semilattices with a selection function, prove a categorical duality for the algebraic semantics of the logic, and use this to derive completeness. We then consider the extension of sub-sub-intuitionistic logic with a variety of axioms.
{"title":"Sub-sub-intuitionistic logic","authors":"Jonte Deakin, Jim de Groot","doi":"arxiv-2408.12030","DOIUrl":"https://doi.org/arxiv-2408.12030","url":null,"abstract":"Sub-sub-intuitionistic logic is obtained from intuitionistic logic by\u0000weakening the implication and removing distributivity. It can alternatively be\u0000viewed as conditional weak positive logic. We provide semantics for\u0000sub-sub-intuitionistic logic by means of semilattices with a selection\u0000function, prove a categorical duality for the algebraic semantics of the logic,\u0000and use this to derive completeness. We then consider the extension of\u0000sub-sub-intuitionistic logic with a variety of axioms.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187779","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}
David Belanger, Chi Tat Chong, Rupert Hölzl, Frank Stephan
We continue the project of the study of reverse mathematics principles inspired by cardinal invariants. In this article in particular we focus on principles encapsulating the existence of large families of objects that are in some sense mutually independent. More precisely, we study the principle MAD stating that a maximal family of pairwise almost disjoint sets exists; and the principle MED expressing the existence of a maximal family of functions that are pairwise eventually different. We investigate characterisations of and relations between these principles and some of their variants. It will turn out that induction strength is an essential parameter in this context.
我们将继续研究受心算不变式启发的逆向数学原理。在这篇文章中，我们特别关注那些概括了在某种意义上相互独立的对象大家族存在性的原理。更确切地说，我们研究了表示存在成对几乎不相交集合的最大族的 MAD 原则；以及表示存在成对最终不同的函数的最大族的 MED 原则。我们研究了这些原理及其一些变体的特征和它们之间的关系。事实将证明，归纳强度是这方面的一个重要参数。
{"title":"Independence and Induction in Reverse Mathematics","authors":"David Belanger, Chi Tat Chong, Rupert Hölzl, Frank Stephan","doi":"arxiv-2408.09796","DOIUrl":"https://doi.org/arxiv-2408.09796","url":null,"abstract":"We continue the project of the study of reverse mathematics principles\u0000inspired by cardinal invariants. In this article in particular we focus on\u0000principles encapsulating the existence of large families of objects that are in\u0000some sense mutually independent. More precisely, we study the principle MAD\u0000stating that a maximal family of pairwise almost disjoint sets exists; and the\u0000principle MED expressing the existence of a maximal family of functions that\u0000are pairwise eventually different. We investigate characterisations of and\u0000relations between these principles and some of their variants. It will turn out\u0000that induction strength is an essential parameter in this context.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142188027","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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