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Uncountable sets and an infinite linear order game 不可数集与无限线性秩序博弈

arXiv - MATH - Logic

Pub Date : 2024-08-26 DOI: arxiv-2408.14624

Tonatiuh Matos-Wiederhold, Luciano Salvetti

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 canhelp characterize countable subsets of the reals. This question is in a similarspirit to how the Banach-Mazur Game characterizes meager sets in an arbitrarytopological 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 tohave a winning strategy on some uncountable set. To this we give a positive answer and moreover construct, for every infinitecardinal $kappa$, a dense linear order of size $kappa$ on which Player II hasa winning strategy on all payoff sets. We finish with some future researchquestions, further underlining the difficulty in generalizing thecharacterization of Brian and Clontz to linear orders.

在马修-贝克（Matthew Baker）的著作[《数学杂志》（Math. Mag. 80 (2007)，第5期，第377-380页]中，出现了一个关于实数集的无限博弈，他问这个博弈能否帮助描述实数的可数子集。这个问题与巴拿赫-马祖尔博弈（Banach-Mazur Game）如何表征任意拓扑空间中的微小集合有着异曲同工之妙。在最近的一篇论文中，威尔-布赖恩（Will Brian）和史蒂文-克隆兹（Steven Clontz）证明，在贝克博弈中，当且仅当报酬集是可数集时，玩家二才有获胜策略。对此，我们给出了肯定的答案，而且，对于每一个无穷心$kappa$，我们都构造了一个大小为$kappa$的密集线性阶，在这个线性阶上，玩家二在所有报酬集上都有获胜策略。最后，我们提出了一些未来的研究问题，进一步强调了将布赖恩和克隆兹的描述推广到线性阶的难度。

引用次数: 0

Rule-Elimination Theorems 规则终结定理

arXiv - MATH - Logic

Pub Date : 2024-08-26 DOI: arxiv-2408.14581

Sayantan Roy

Cut-elimination theorems constitute one of the most important classes oftheorems of proof theory. Since Gentzen's proof of the cut-elimination theoremfor the system $mathbf{LK}$, several other proofs have been proposed. Eventhough the techniques of these proofs can be modified to sequent systems otherthan $mathbf{LK}$, they are essentially of a very particular nature; each ofthem describes an algorithm to transform a given proof to a cut-free proof.However, due to its reliance on heavy syntactic arguments and casedistinctions, such an algorithm makes the fundamental structure of the argumentrather opaque. We, therefore, consider rules abstractly, within the frameworkof 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 dothis, we first give a non-algorithmic proof of the cut-elimination theorem forthe propositional fragment of $mathbf{LK}$. From this proof, we abstract theessential features of the argument and define something called ``normal sequentstructures'' relative to a particular rule. We then prove a version of therule-elimination theorem for these. Abstracting even more, we define ``abstractsequent structures'' and show that for these structures, the correspondingversion of the ``rule''-elimination theorem has a converse as well.

割除定理是证明理论中最重要的定理之一。自从根岑证明了$mathbf{LK}$系统的割除定理以来，人们又提出了其他一些证明。尽管这些证明的技术可以被修改用于 $mathbf{LK}$ 以外的序列系统，但它们本质上都具有非常特殊的性质；它们中的每一个都描述了一种将给定证明转换为无剪切证明的算法。然而，由于这种算法依赖于大量的句法论证和分例，它使得论证者的基本结构变得不透明。因此，我们在普遍逻辑所熟悉的逻辑结构框架内抽象地考虑规则，旨在阐明所谓 "消除定理 "的本质。为此，我们首先给出了$mathbf{LK}$命题片段的割除定理的非算法证明。从这个证明中，我们抽象出论证的基本特征，并定义了相对于特定规则的 "正常序列结构"。然后，我们为这些规则证明一个版本的规则消除定理。为了进一步抽象，我们定义了 "抽象后序结构"，并证明对于这些结构，"规则"-消除定理的相应版本也有一个逆定理。

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引用次数: 0

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