Annals of Pure and Applied Logic最新文献

英文中文

On cardinal invariants related to Rosenthal families and large-scale topology 关于Rosenthal族与大尺度拓扑的基数不变量

IF 0.6 2区数学 Q2 LOGIC

Annals of Pure and Applied Logic

Pub Date : 2025-08-01 Epub Date: 2025-05-05 DOI: 10.1016/j.apal.2025.103607

Arturo Martínez-Celis, Tomasz Żuchowski

Given a function

f \in ω^{ω}

, a set

A \in {[ω]}^{ω}

is free for f if

f [A] \cap A

is finite. For a class of functions

Γ \subseteq ω^{ω}

, we define

{ros}_{Γ}

as the smallest size of a family

A \subseteq {[ω]}^{ω}

such that for every

f \in Γ

there is a set

A \in A

which is free for f, and

Δ_{Γ}

as the smallest size of a family

F \subseteq Γ

such that for every

A \in {[ω]}^{ω}

there is

f \in F

such that A is not free for f. We compare several versions of these cardinal invariants with some of the classical cardinal characteristics of the continuum. Using these notions, we partially answer some questions from [20] and [2].

给定一个函数f∈ω，如果f[a]∩a是有限的，则集合a∈[ω]ω对于f是自由的。对于一类函数Γ≥ω，我们将rosΓ定义为族a的最小规模，使得对于每一个f∈Γ，存在一个对f自由的集合a∈a；将ΔΓ定义为族f的最小规模，使得对于每一个a∈[ω]ω，存在f∈f，使得a对f不自由。我们将这些基本不变量的几种版本与连续体的一些经典基本特征进行比较。利用这些概念，我们部分地回答了[20]和[2]中的一些问题。

引用次数: 0

Spaces not distinguishing ideal pointwise and σ-uniform convergence 不区分理想点和σ-一致收敛的空间

IF 0.6 2区数学 Q2 LOGIC

Annals of Pure and Applied Logic

Pub Date : 2025-08-01 Epub Date: 2025-05-08 DOI: 10.1016/j.apal.2025.103609

Rafał Filipów, Adam Kwela

We examine topological spaces not distinguishing ideal pointwise and ideal σ-uniform convergence of sequences of real-valued continuous functions defined on them. For instance, we introduce a purely combinatorial cardinal characteristic (a sort of the bounding number

b

) and prove that it describes the minimal cardinality of topological spaces which distinguish ideal pointwise and ideal σ-uniform convergence. Moreover, we provide examples of topological spaces (focusing on subsets of reals) that do or do not distinguish the considered convergences. Since similar investigations for ideal quasi-normal convergence instead of ideal σ-uniform convergence have been performed in literature, we also study spaces not distinguishing ideal quasi-normal and ideal σ-uniform convergence of sequences of real-valued continuous functions defined on them.

研究了不区分在其上定义的实值连续函数序列的理想点和理想σ-一致收敛的拓扑空间。例如，我们引入了一个纯组合基数特征（一种边界数b），并证明了它描述了区分理想点向和理想σ-一致收敛的拓扑空间的最小基数。此外，我们还提供了拓扑空间的示例（关注实数的子集），它们区分或不区分所考虑的收敛性。由于文献中对理想拟正规收敛代替理想σ-一致收敛进行了类似的研究，我们也研究了在其上定义的实值连续函数序列的不区分理想拟正规收敛和理想σ-一致收敛的空间。

引用次数: 0

Generically computable linear orderings 一般可计算的线性排序

IF 0.6 2区数学 Q2 LOGIC

Annals of Pure and Applied Logic

Pub Date : 2025-08-01 Epub Date: 2025-05-12 DOI: 10.1016/j.apal.2025.103612

Wesley Calvert , Douglas Cenzer , David Gonzalez , Valentina Harizanov

We study notions of generic and coarse computability in the context of computable structure theory. Our notions are stratified by the

Σ_{β}

hierarchy. We focus on linear orderings. We show that at the

Σ_{1}

level, all linear orderings have both generically and coarsely computable copies. This behavior changes abruptly at higher levels; we show that at the

Σ_{α + 2}

level for any

α \in ω_{1}^{C K}

the set of linear orderings with generically or coarsely computable copies is

Σ_{1}^{1}

-complete and therefore maximally complicated. This development is new even in the general analysis of generic and coarse computability of countable structures. In the process of proving these results, we introduce new tools for understanding generically and coarsely computable structures. We are able to give a purely structural statement that is equivalent to having a generically computable copy and show that every relational structure with only finitely many relations has coarsely and generically computable copies at the lowest level of the hierarchy.

在可计算结构理论的背景下，研究了一般可计算性和粗可计算性的概念。我们的观念是由Σβ层次结构分层的。我们关注的是线性排序。我们证明了在Σ1水平上，所有的线性排序都具有一般和粗可计算的副本。这种行为在更高的层次上突然改变；我们证明了在Σα+2水平上，对于任意α∈ω1CK，具有一般或粗可计算副本的线性排序集是Σ11-complete，因此是最复杂的。这一发展在可数结构的一般可计算性和粗可计算性的一般分析中也是新的。在证明这些结果的过程中，我们引入了新的工具来理解一般和粗计算结构。我们能够给出一个等价于具有一般可计算副本的纯结构命题，并证明每个只有有限多个关系的关系结构在层次结构的最低层次上都具有粗糙且一般可计算的副本。

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引用次数: 0

Metric spaces in choiceless set theory 无选择集合论中的度量空间

IF 0.6 2区数学 Q2 LOGIC

Annals of Pure and Applied Logic

Pub Date : 2025-08-01 Epub Date: 2025-04-22 DOI: 10.1016/j.apal.2025.103603

Eleftherios Tachtsis

<div><div>We answer open questions from Keremedis (2016) [12] and Keremedis and Tachtsis (2022) [16], and properly strengthen some results from the above papers as well as from Keremedis et al. (2023) [19]. In particular, and among other results, we establish the following:<ul><li>1.<div>The Boolean Prime Ideal Theorem does not imply “For every sequentially compact metric space <math><mo>〈</mo><mi>X</mi><mo>,</mo><mi>d</mi><mo>〉</mo></math>, <math><mo>|</mo><mi>X</mi><mo>|</mo><mo>≤</mo><msup><mrow><mn>2</mn></mrow><mrow><msub><mrow><mi>ℵ</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></msup></math>” in ZF (Zermelo–Fraenkel set theory without the Axiom of Choice (AC)).</div></li><li>2.<div>“Every linearly ordered set can be well ordered” ∧ “The union of a well-orderable family of well-orderable sets is well orderable” ∧ “For every uncountable sequentially compact metric space <math><mo>〈</mo><mi>X</mi><mo>,</mo><mi>d</mi><mo>〉</mo></math>, <math><mo>|</mo><mi>X</mi><mo>|</mo><mo>=</mo><msup><mrow><mn>2</mn></mrow><mrow><msub><mrow><mi>ℵ</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></msup></math>” does not imply the Axiom of Countable Choice in ZFA (ZF with atoms).</div></li><li>3.<div>“For every uncountable compact metric space <math><mo>〈</mo><mi>X</mi><mo>,</mo><mi>d</mi><mo>〉</mo></math>, <math><mo>|</mo><mi>X</mi><mo>|</mo><mo>≥</mo><msup><mrow><mn>2</mn></mrow><mrow><msub><mrow><mi>ℵ</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></msup></math>” does not imply “For every uncountable sequentially compact metric space <math><mo>〈</mo><mi>X</mi><mo>,</mo><mi>d</mi><mo>〉</mo></math>, <math><mo>|</mo><mi>X</mi><mo>|</mo><mo>≥</mo><msup><mrow><mn>2</mn></mrow><mrow><msub><mrow><mi>ℵ</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></msup></math>” in ZFA”.</div></li><li>4.<div>“For every uncountable sequentially compact metric space <math><mo>〈</mo><mi>X</mi><mo>,</mo><mi>d</mi><mo>〉</mo></math> <math><mo>|</mo><mi>X</mi><mo>|</mo><mo>≥</mo><msup><mrow><mn>2</mn></mrow><mrow><msub><mrow><mi>ℵ</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></msup></math>” does not imply “For every uncountable compact metric space <math><mo>〈</mo><mi>X</mi><mo>,</mo><mi>d</mi><mo>〉</mo></math>, <math><mo>|</mo><mi>X</mi><mo>|</mo><mo>=</mo><msup><mrow><mn>2</mn></mrow><mrow><msub><mrow><mi>ℵ</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></msup></math>” in ZFA.</div></li><li>5.<div>“For every uncountable sequentially compact metric space <math><mo>〈</mo><mi>X</mi><mo>,</mo><mi>d</mi><mo>

我们回答了Keremedis(2016)[12]和Keremedis and Tachtsis(2022)[19]的开放性问题，并适当加强了上述论文以及Keremedis等人（2023）[19]的一些结果。特别地，在其他结果中，我们确立了以下几点：在ZF （Zermelo-Fraenkel集合论，无选择公理（AC））.2中，布尔素数理想定理并不意味着“对于每一个序紧度量空间< X,d >, |X|≤2¹0”。“每个线性有序集都可以良序”∧良序集族的并是良序的“∧”对于每一个不可数序紧度量空间< X,d >, |X|=2¹0”并不意味着ZFA（含原子的ZF）中的可数选择公理。“对于每一个不可数紧度量空间< X,d >, |X|≥2¹0”并不意味着在ZFA中“对于每一个不可数连续紧度量空间< X,d >, |X|≥2¹0”。ZFA.5中的“对于每一个不可数连续紧度量空间< X,d > |X|≥2¹0”并不意味着“对于每一个不可数紧度量空间< X,d >, |X|=2¹0”。“对于每一个不可数序紧度量空间< X,d > |X|≥2 ^ 0”并不蕴涵ZFA.6中可数集的可数选择公理（CMCω）。“每一个线性有序的集合都是良序的”并不意味着ZFA中的“有限集合的可数选择公理”∨“每一个无限紧度量空间都有一个无限分散的子空间”。我们还讨论了斯通定理“每个度量空间都是准紧的”（ST）的演绎强度的开放问题，并提供了ST是否意味着CMCω的非平凡部分答案。特别地，我们证明了形式上较弱的命题“对于每一个度量空间< X,d >， |X|≥2 ^ 0或< X,d >是准紧的”并不意味着在ZFA中有CMCω。我们还证明，对于每一个不可数正则基数κ，在ZF中的“for all infinite良序基数λ<；κ， the Principle of Dependent Choices for λ成立”并不隐含上述ST的弱形式。

引用次数: 0

Weakly o-minimal types 弱o极小型

IF 0.6 2区数学 Q2 LOGIC

Annals of Pure and Applied Logic

Pub Date : 2025-08-01 Epub Date: 2025-05-02 DOI: 10.1016/j.apal.2025.103605

Slavko Moconja , Predrag Tanović

We introduce and study weak o-minimality in the context of complete types in an arbitrary first-order theory. A type

p \in S (A)

is weakly o-minimal if for some relatively A-definable linear order, <, on

p (C)

every relatively

L_{C}

-definable subset of

p (C)

has finitely many convex components in

(p (C), <)

. We establish many nice properties of weakly o-minimal types. For example, we prove that weakly o-minimal types are dp-minimal and share several properties of weight-one types in stable theories, and that a version of monotonicity theorem holds for relatively definable functions on the locus of a weakly o-minimal type.

在任意一阶理论中引入并研究了完全类型下的弱极小性。如果在p(C)上p∈S(A)是弱o-极小型的，对于某个相对A-可定义的线性序列<；， p(C)上p(C)的每个相对lc -可定义的子集在（p(C),<）中有有限多个凸分量。我们建立了弱o极小型的许多很好的性质。例如，我们证明了弱o-极小型是dp-极小型，并且在稳定理论中具有若干权重- 1型的性质，并且证明了在弱o-极小型的轨迹上相对可定义函数的单调性定理的一个版本成立。

引用次数: 0

Upward Löwenheim-Skolem-Tarski numbers for abstract logics 向上Löwenheim-Skolem-Tarski数字抽象逻辑

IF 0.6 2区数学 Q2 LOGIC

Annals of Pure and Applied Logic

Pub Date : 2025-08-01 Epub Date: 2025-03-19 DOI: 10.1016/j.apal.2025.103583

Victoria Gitman , Jonathan Osinski

Galeotti, Khomskii and Väänänen recently introduced the notion of the upward Löwenheim-Skolem-Tarski number for a logic, strengthening the classical notion of a Hanf number. A cardinal κ is the upward Löwenheim-Skolem-Tarski number (ULST number) of a logic

L

if it is the least cardinal with the property that whenever M is a model of size at least κ satisfying a sentence φ in

L

, then there are arbitrarily large models satisfying φ and having M as a substructure. The substructure requirement is what differentiates the ULST number from the Hanf number and gives the notion large cardinal strength. While it is a theorem of ZFC that every logic has a Hanf number, Galeotti, Khomskii and Väänänen showed that the existence of the ULST number for second-order logic implies the existence of a partially extendible cardinal. We answer positively their conjecture that the ULST number for second-order logic is the least extendible cardinal.

We define the strong ULST number by strengthening the substructure requirement to elementary substructure. We investigate the ULST and strong ULST numbers for several classical strong logics: infinitary logics, the equicardinality logic, logic with the well-foundedness quantifier, second-order logic, and sort logics. We show that the ULST and the strong ULST numbers are characterized in some cases by classical large cardinals and in some cases by natural new large cardinal notions that they give rise to. We show that for some logics the notions of the ULST number, strong ULST number and least strong compactness cardinal coincide, while for others, it is consistent that they can be separated. Finally, we introduce a natural large cardinal notion characterizing strong compactness cardinals for the equicardinality logic.

Galeotti， Khomskii和Väänänen最近为逻辑引入了向上Löwenheim-Skolem-Tarski数的概念，加强了汉夫数的经典概念。基数κ是逻辑L的向上Löwenheim-Skolem-Tarski数（ULST数），如果它是具有以下性质的最小基数：每当M是大小至少κ的模型满足L中的句子φ时，则存在满足φ且M为子结构的任意大模型。子结构要求是ULST数与汉夫数的区别，并赋予了大基数强度的概念。虽然每个逻辑都有一个汉夫数是ZFC的一个定理，但Galeotti、Khomskii和Väänänen证明了二阶逻辑的ULST数的存在意味着部分可扩展基数的存在。我们肯定地回答了他们关于二阶逻辑的ULST数是最小可扩展基数的猜想。我们通过将子结构要求强化为基本子结构来定义强ULST数。我们研究了几种经典强逻辑的ULST和强ULST数：无限逻辑、等价逻辑、良基量词逻辑、二阶逻辑和排序逻辑。我们表明，在某些情况下，ULST和强ULST数的特征是经典的大基数，而在某些情况下，特征是由它们产生的自然的新的大基数概念。我们证明了在某些逻辑中，ULST数、强ULST数和最弱紧性的概念基本重合，而在另一些逻辑中，它们可以分开是一致的。最后，我们引入了一个自然大基数概念，表征了等价逻辑的强紧性基数。

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引用次数: 0

The power of the Binary Value Principle 二进制值原理的力量

IF 0.6 2区数学 Q2 LOGIC

Annals of Pure and Applied Logic

Pub Date : 2025-08-01 Epub Date: 2025-05-21 DOI: 10.1016/j.apal.2025.103614

Yaroslav Alekseev , Edward A. Hirsch

The (extended) Binary Value Principle (

eBVP

, the equation

\sum_{i = 1}^{n} x_{i} 2^{i - 1} = - k

for

k > 0

and Boolean variables

x_{i}

) has received a lot of attention recently, several lower bounds have been proved for it [1], [2], [11]. Also it has been shown [1] that the probabilistically verifiable Ideal Proof System (

IPS

) [8] together with

eBVP

polynomially simulates a similar semialgebraic proof system. In this paper we consider Polynomial Calculus with an algebraic version of Tseitin's extension rule (

Ext - PC

) that introduces a new variable for any polynomial. Contrary to

IPS

, this is a Cook–Reckhow proof system. We show that in this context

eBVP

still allows to simulate similar semialgebraic systems. We also prove that it allows to simulate the Square Root Rule [6], which is in sharp contrast with the result of [2] that shows an exponential lower bound on the size of

Ext - PC

derivations of the Binary Value Principle from its square. On the other hand, we demonstrate that

eBVP

probably does not help in proving exponential lower bounds for Boolean formulas: we show that an

Ext - PC

(even with the Square Root Rule) derivation of any unsatisfiable Boolean formula in CNF from

eBVP

must be of exponential size.

（扩展的）二元值原理（eBVP，方程∑i=1nxi2i−1= - k对于k>；0和布尔变量xi）最近受到了很多关注，已经证明了它的几个下界[1]，[2],[11]。并且证明了概率可验证的理想证明系统（IPS）[8]与eBVP一起多项式地模拟了一个类似的半代数证明系统。本文用tseittin扩展规则（Ext-PC）的代数版本来考虑多项式微积分，该扩展规则为任意多项式引入了一个新变量。与IPS相反，这是一个库克-清算证明系统。我们表明，在这种情况下，eBVP仍然允许模拟类似的半代数系统。我们还证明了它允许模拟平方根规则[6]，这与[2]的结果形成鲜明对比，[2]的结果显示了二进制值原理从其平方导出的Ext-PC的大小的指数下界。另一方面，我们证明了eBVP可能对证明布尔公式的指数下界没有帮助：我们证明了从eBVP推导出CNF中任何不满足的布尔公式的Ext-PC（即使使用平方根规则）必须具有指数大小。

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引用次数: 0

The singleton degrees of the Σ20 sets are not dense Σ20集合的单态度是不密集的

IF 0.6 2区数学 Q2 LOGIC

Annals of Pure and Applied Logic

Pub Date : 2025-08-01 Epub Date: 2025-05-27 DOI: 10.1016/j.apal.2025.103616

Thomas F. Kent , Keng Meng Ng , Andrea Sorbi

Answering an open question raised by Cooper, we show that there exist

Δ_{2}^{0}

sets D and E such that the singleton degree of E is a minimal cover of the singleton degree of D. This shows that the

Σ_{2}^{0}

singleton degrees, and the

Δ_{2}^{0}

singleton degrees, are not dense (and consequently the

Π_{2}^{0}

Q-degrees, and the

Δ_{2}^{0}

Q-degrees, are not dense). Moreover, D and E can be built to lie in the same enumeration degree.

回答Cooper提出的一个开放性问题，我们证明存在Δ20集合D和E，使得E的单态度是D的单态度的最小覆盖。这表明Σ20单态度和Δ20单态度不是密集的（因此Π20 q度和Δ20 q度不是密集的）。而且，D和E可以构建在同一枚举度。

引用次数: 0

Model-theoretic K1 of free modules over PIDs pid上自由模的模型论K1

IF 0.6 2区数学 Q2 LOGIC

Annals of Pure and Applied Logic

Pub Date : 2025-08-01 Epub Date: 2025-05-22 DOI: 10.1016/j.apal.2025.103613

Sourayan Banerjee, Amit Kuber

Motivated by Krajiček and Scanlon's definition of the Grothendieck ring

K_{0} (M)

of a first-order structure M, we introduce the definition of K-groups

K_{n} (M)

for

n \geq 0

via Quillen's

S^{- 1} S

construction. We provide a recipe for the computation of

K_{1} (M_{R})

, where

M_{R}

is a free module over a PID R, subject to the knowledge of the abelianizations of the general linear groups

G L_{n} (R)

. As a consequence, we provide explicit computations of

K_{1} (M_{R})

when R belongs to a large class of Euclidean domains that includes fields with at least 3 elements and polynomial rings over fields with characteristic 0. We also show that the algebraic

K_{1}

of a PID R embeds into

K_{1} (R_{R})

基于krajiekk和Scanlon关于一阶结构M的Grothendieck环K0(M)的定义，我们通过Quillen的S−1S构造引入了n≥0时k群Kn(M)的定义。我们提供了K1（MR）的计算公式，其中MR是PID R上的自由模，受一般线性群GLn(R)的阿贝尔化的知识的约束。因此，我们提供了K1（MR）的显式计算，当R属于欧几里得域的一大类，其中包括至少有3个元素的域和特征为0的域上的多项式环。我们还证明了PID R的代数K1嵌入到K1（RR）中。

引用次数: 0

Free p-algebras revisited: An algebraic investigation of implication-free intuitionism 重访自由p-代数：无蕴涵直觉主义的代数研究

IF 0.6 2区数学 Q2 LOGIC

Annals of Pure and Applied Logic

Pub Date : 2025-08-01 Epub Date: 2025-05-08 DOI: 10.1016/j.apal.2025.103610

Tomasz Kowalski , Katarzyna Słomczyńska

We give a new construction of free distributive p-algebras. Our construction relies on a detailed description of completely meet-irreducible congruences, so it is purely universal algebraic. It yields a normal form theorem for p-algebra terms, simpler proofs of several existing results. As a by-product, we obtain an isomorphism between the free pseudocomplemented semilattice and the poset of join-irreducibles of the free p-algebra augmented by zero.

给出了自由分配p-代数的一个新构造。我们的构造依赖于对完全满足不可约同余的详细描述，因此它是纯粹的普适代数。它给出了p代数项的一个标准形式定理，对几个已有结果的更简单的证明。作为副产物，我们得到了自由伪补半格与自由增0 p代数的联合不可约正集之间的同构。

引用次数: 0

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