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Groups definable in Presburger arithmetic 可在普氏算术中定义的群
IF 0.6 2区 数学 Q2 LOGIC Pub Date : 2024-08-10 DOI: 10.1016/j.apal.2024.103507
Juan Pablo Acosta

Here we give a complete list of the groups definable in Presburger arithmetic up to a finite index subgroup.

在此,我们将给出一份完整的列表,列出可在普雷斯伯格算术中定义的群,直至一个有限索引子群。
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引用次数: 0
A complete axiomatization of infinitary first-order intuitionistic logic over Lκ+,κ 无穷一阶直观逻辑在[公式省略]上的完整公理化
IF 0.6 2区 数学 Q2 LOGIC Pub Date : 2024-08-08 DOI: 10.1016/j.apal.2024.103506
Christian Espíndola

Given a weakly compact cardinal κ, we give an axiomatization of intuitionistic first-order logic over Lκ+,κ and prove it is sound and complete with respect to Kripke models. As a consequence we get the disjunction and existence properties for that logic. This generalizes the work of Nadel in [8] for intuitionistic logic over Lω1,ω. When κ is a regular cardinal such that κ<κ=κ, we deduce, by an easy modification of the proof, a complete axiomatization of intuitionistic first-order logic over Lκ+,κ,κ, the language with disjunctions of at most κ formulas, conjunctions of less than κ formulas and quantification on less than κ many variables. In particular, this applies to any regular cardinal under the Generalized Continuum Hypothesis.

给定一个弱紧凑红心κ,我们给出了Lκ+,κ上的直观一阶逻辑的公理化,并证明它在克里普克模型方面是健全和完备的。因此,我们得到了该逻辑的析取和存在性质。这概括了纳德尔在 [8] 中针对 Lω1,ω 上的直觉逻辑所做的工作。当κ是一个正则红心数,使得κ<κ=κ时,我们通过对证明的简单修改,推导出了Lκ+,κ,κ上的直观一阶逻辑的完整公理化,这种语言具有最多κ个公式的分结、少于κ个公式的连接和少于κ个变量的量化。这尤其适用于广义连续假说下的任何正则心项。
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引用次数: 0
μ-clubs of Pκ(λ): Paradise in heaven μ-clubs of Pκ(λ):天堂中的天堂
IF 0.6 2区 数学 Q2 LOGIC Pub Date : 2024-07-24 DOI: 10.1016/j.apal.2024.103497
Pierre Matet

Let μ<κ<λ be three infinite cardinals, the first two being regular. We show that if there is no inner model with large cardinals, u(κ,λ) is regular, where u(κ,λ) denotes the least size of a cofinal subset in (Pκ(λ),), and cf(λ)μ, then (a) the μ-club filters on Pκ(λ) and Pκ(u(κ,λ)) are isomorphic, and (b) the ideal dual to the μ-club filter on Pκ(λ) (and hence the restriction of the nonstationary ideal on Pκ(λ) to sets of uniform cofinality μ) is not Iκ,λ-bu(κ,λ)-saturated.

设μ<κ<λ是三个无限红心,前两个是正则。我们证明,如果不存在具有大红心的内部模型,u(κ,λ) 是正则的,其中 u(κ,λ) 表示 (Pκ(λ),⊆) 中的最小同尾子集,且 cf(λ)≠μ,那么 (a) Pκ(λ) 和 Pκ(u(κ. λ)) 上的μ-club 过滤器是同构的、λ)上的 μ-club 过滤器是同构的,以及 (b) Pκ(λ) 上 μ-club 过滤器的对偶理想(因此 Pκ(λ) 上的非稳态理想对均匀同系数 μ 集合的限制)不是 Iκ,λ-bu(κ,λ) 饱和的。
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引用次数: 0
First-order reasoning and efficient semi-algebraic proofs 一阶推理和高效半代数证明
IF 0.6 2区 数学 Q2 LOGIC Pub Date : 2024-07-14 DOI: 10.1016/j.apal.2024.103496
Fedor Part , Neil Thapen , Iddo Tzameret

Semi-algebraic proof systems such as sum-of-squares (SoS) have attracted a lot of attention due to their relation to approximation algorithms: constant degree semi-algebraic proofs lead to conjecturally optimal polynomial-time approximation algorithms for important NP-hard optimization problems. Motivated by the need to allow a more streamlined and uniform framework for working with SoS proofs than the restrictive propositional level, we initiate a systematic first-order logical investigation into the kinds of reasoning possible in algebraic and semi-algebraic proof systems. Specifically, we develop first-order theories that capture in a precise manner constant degree algebraic and semi-algebraic proof systems: every statement of a certain form that is provable in our theories translates into a family of constant degree polynomial calculus or SoS refutations, respectively; and using a reflection principle, the converse also holds.

This places algebraic and semi-algebraic proof systems in the established framework of bounded arithmetic, while providing theories corresponding to systems that vary quite substantially from the usual propositional-logic ones.

We give examples of how our semi-algebraic theory proves statements such as the pigeonhole principle, we provide a separation between algebraic and semi-algebraic theories, and we describe initial attempts to go beyond these theories by introducing extensions that use the inequality symbol, identifying along the way which extensions lead outside the scope of constant degree SoS. Moreover, we prove new results for propositional proofs, and specifically extend Berkholz's dynamic-by-static simulation of polynomial calculus (PC) by SoS to PC with the radical rule.

半代数证明系统,如平方和(),因其与近似算法的关系而备受关注:常度半代数证明为重要的困难优化问题带来了猜想中最优的多项式时间近似算法。与限制性的命题层面相比,我们需要一个更精简、更统一的框架来处理证明,受此激励,我们开始对代数和半代数证明系统中可能的推理类型进行系统的一阶逻辑研究。具体地说,我们发展了一阶理论,以精确的方式捕捉常度代数和半代数证明系统:在我们的理论中,每一个可证明的特定形式的陈述都分别转化为一系列常度多项式微积分或反驳;利用反射原理,反过来也成立。
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引用次数: 0
Owings-like theorems for infinitely many colours or finite monochromatic sets 无限多色或有限单色集的类欧文定理
IF 0.6 2区 数学 Q2 LOGIC Pub Date : 2024-07-10 DOI: 10.1016/j.apal.2024.103495
David J. Fernández-Bretón, Eliseo Sarmiento Rosales, Germán Vera

Inspired by Owings's problem, we investigate whether, for a given an Abelian group G and cardinal numbers κ,θ, every colouring c:Gθ yields a subset XG with |X|=κ such that X+X is monochromatic. (Owings's problem asks this for G=Z, θ=2 and κ=0; this is known to be false for the same G and κ but θ=3.) We completely settle the question for κ and θ both finite (by obtaining sufficient and necessary conditions for a positive answer) and for κ and θ both infinite (with a negative answer). Also, in the case where θ is infinite but κ is finite, we obtain some sufficient conditions for a negative answer as well as an example with a positive answer.

受欧文斯问题的启发,我们研究了对于给定的阿贝尔群和心数 ,每一种着色是否都能产生一个子集,而这个子集又是单色的。(欧文斯的问题是针对 、 和 提出这个问题的;众所周知,对于相同的 和 ,这个问题是假的)。对于 和 都是有限的(通过得到肯定答案的充分必要条件),以及对于 和 都是无限的(否定答案),我们完全解决了这个问题。此外,在 和 是无限的,但 和 是有限的情况下,我们还得到了一些得到否定答案的充分条件,以及一个得到肯定答案的例子。
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引用次数: 0
Computably and punctually universal spaces 可计算和准时通用空间
IF 0.6 2区 数学 Q2 LOGIC Pub Date : 2024-07-10 DOI: 10.1016/j.apal.2024.103491
Ramil Bagaviev , Ilnur I. Batyrshin , Nikolay Bazhenov , Dmitry Bushtets , Marina Dorzhieva , Heer Tern Koh , Ruslan Kornev , Alexander G. Melnikov , Keng Meng Ng

We prove that the standard computable presentation of the space C[0,1] of continuous real-valued functions on the unit interval is computably and punctually (primitively recursively) universal. From the perspective of modern computability theory, this settles a problem raised by Sierpiński in the 1940s.

We prove that the original Urysohn's construction of the universal separable Polish space U is punctually universal. We also show that effectively compact, punctual Stone spaces are punctually homeomorphically embeddable into Cantor space 2ω; note that we do not require effective compactness be primitive recursive. We also prove that effective compactness cannot be dropped from the premises by constructing a counterexample.

我们证明了单位区间上连续实值函数空间 C[0,1] 的标准可计算性呈现是可计算的、准时的(原始递归的)普适的。从现代可计算性理论的角度来看,这解决了西尔潘斯基在 20 世纪 40 年代提出的一个问题。我们证明了最初乌里索恩构造的波兰通用可分离空间 U 是准时通用的。我们还证明了有效紧凑的、准时的斯通空间是准时同构地嵌入到康托空间 2ω中的;注意,我们并不要求有效紧凑性是原始递归的。我们还通过构造一个反例证明了有效紧凑性不能从前提中丢弃。
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引用次数: 0
Concerning Keisler measures over ultraproducts 关于超积物上的凯斯勒量纲
IF 0.6 2区 数学 Q2 LOGIC Pub Date : 2024-07-09 DOI: 10.1016/j.apal.2024.103492
Kyle Gannon

As consequence of the VC theorem, any pseudo-finite measure over an NIP ultraproduct is generically stable. We demonstrate a converse of this theorem and prove that any finitely approximable measure over an ultraproduct is itself pseudo-finite (even without the NIP assumption). We also analyze the connection between the Morley product and the pseudo-finite product. In particular, we show that if μ is definable and both μ and ν are pseudo-finite, then the Morley product of μ and ν agrees with the pseudo-finite product of μ and ν. Using this observation, we construct generically stable idempotent measures on pseudo-finite NIP groups.

根据 VC 定理,NIP 超积上的任何伪有限度量都是一般稳定的。我们证明了这一定理的逆定理,并证明超积上的任何有限可近似度量本身都是伪无限的(即使没有 NIP 假设)。我们还分析了莫利积和伪无限积之间的联系。特别是,我们证明,如果是可定义的,并且 和 都是伪无限的,那么 和 的莫利积与 和 的伪无限积是一致的。利用这一观察结果,我们构建了伪无限 NIP 群上一般稳定的幂级数。
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引用次数: 0
Elimination of quantifiers for a theory of real closed rings 实闭环理论的量词消除
IF 0.6 2区 数学 Q2 LOGIC Pub Date : 2024-07-06 DOI: 10.1016/j.apal.2024.103494
Jorge I. Guier

Let T be the theory of lattice-ordered, convex subrings of von Neumann regular real closed rings that are divisible-projectable, sc-regular ([12]) and have no minimal (non zero) idempotents. In this paper, we introduce and study a local divisibility binary relation that, added to the language for lattice-ordered rings, together with the (usual) divisibility relation and the radical relation associated to the minimal prime spectrum ([19]) yields quantifier elimination for T.

让 T⁎ 成为冯-诺依曼正则实闭环的格有序凸子环理论,这些子环是可分投影的、sc 正则的([12]),并且没有最小(非零)幂等子。在本文中,我们引入并研究了一种局部可分性二元关系,这种关系与(通常的)可分关系以及与最小素谱相关的基元关系([19])一起加入到格有序环的语言中,产生了 T⁎ 的量词消元。
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引用次数: 0
Model-completeness and decidability of the additive structure of integers expanded with a function for a Beatty sequence 用一个函数扩展的整数加法结构的模型完备性和可判定性的比蒂序列
IF 0.6 2区 数学 Q2 LOGIC Pub Date : 2024-07-06 DOI: 10.1016/j.apal.2024.103493
Mohsen Khani , Ali N. Valizadeh , Afshin Zarei

We introduce a model-complete theory which completely axiomatizes the structure Zα=Z,+,0,1,f where f:xαx is a unary function with α a fixed transcendental number. Moreover, we show that decidability of Zα is equivalent to computability of α. This result fits into the more general theme of adding traces of multiplication to integers without losing decidability.

我们引入了一个模型完备的理论,它完全公理化了结构 Zα=〈Z,+,0,1,f〉,其中 f:x↦⌊αx⌋ 是一个一元函数,α 是一个固定的超越数。此外,我们还证明了 Zα 的可判定性等同于 α 的可计算性。这一结果与在不丧失可判定性的前提下为整数添加乘法踪迹这一更普遍的主题相吻合。
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引用次数: 0
Club stationary reflection and other combinatorial principles at ℵω+2 ℵω+2处的俱乐部静止反射和其他组合原理
IF 0.6 2区 数学 Q2 LOGIC Pub Date : 2024-07-01 DOI: 10.1016/j.apal.2024.103489

In this paper we continue the study in [11] of compactness and incompactness principles at double successors, focusing here on the case of double successors of singulars of countable cofinality. We obtain models which satisfy the tree property and club stationary reflection at these double successors. Moreover, we can additionally obtain either approachability or its failure. We also show how to obtain our results on ω+2 by incorporating collapses; particularly relevant for these circumstances is a new indestructibility theorem of ours showing that posets satisfying certain linked assumptions preserve club stationary reflection.

在本文中,我们将继续研究双继点的紧凑性和不紧凑性原则,重点是可数同频奇点的双继点。我们得到了在这些双继点上满足树性质和俱乐部静止反映的模型。此外,我们还能得到可接近性或其失败。我们还展示了如何通过结合坍缩来获得我们的结果;与这些情况特别相关的是我们的一个新的不灭性定理,它表明满足某些关联假设的正集保留了俱乐部静止反映。
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引用次数: 0
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Annals of Pure and Applied Logic
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