Annals of Pure and Applied Logic最新文献

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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 : 2025-05-05 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

Good projective witnesses 好的投影证人

IF 0.6 2区数学 Q2 LOGIC

Annals of Pure and Applied Logic

Pub Date : 2025-05-05 DOI: 10.1016/j.apal.2025.103606

Vera Fischer , Sy David Friedman , David Schrittesser , Asger Törnquist

We develop a new forcing notion for adjoining self-coding cofinitary permutations and use it to show that consistently, the minimal cardinality

a_{g}

of a maximal cofinitary group (MCG) is strictly between

ℵ_{1}

and

c

, and there is a

Π_{2}^{1}

-definable MCG of this cardinality. Here

Π_{2}^{1}

is optimal, making this result a natural counterpart to the Borel MCG of Horowitz and Shelah. Our theorem has its analogue in the realm of maximal almost disjoint (MAD) families, extending a line of results regarding the definability properties of MAD families in models with large continuum.

本文提出了相邻自编码共限置换的一个新的强制概念，并利用它证明了最大共限群（MCG）的最小基数ag严格地存在于λ 1和λ c之间，并且存在这个基数的Π21-definable MCG。这里Π21是最优的，使这个结果与霍洛维茨和希拉的Borel MCG自然对应。我们的定理在极大几乎不相交族（MAD）领域有类似的结果，扩展了关于大连续统模型中极大几乎不相交族的可定义性的一系列结果。

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 : 2025-05-02 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

Unreachability of Γ2n+1,m 无法到达Γ2n+1，m

IF 0.6 2区数学 Q2 LOGIC

Annals of Pure and Applied Logic

Pub Date : 2025-04-23 DOI: 10.1016/j.apal.2025.103604

Derek Levinson

We find bounds for the maximal length of a sequence of distinct

Γ_{2n + 1, m}

-sets under AD and show there is no sequence of distinct

Γ_{2n + 1}

-sets of length

δ_{2n + 3}^{1}

. As a special case, there is no sequence of distinct

Γ_{1, m}

-sets of length

ℵ_{m + 2}

. These are the optimal results for the pointclasses

Γ_{2n + 1}

and

Γ_{1, m}

我们找到了在AD条件下具有不同Γ2n+1，m-集的序列的最大长度的界，并证明不存在具有不同Γ2n+1-集的序列，其长度为δ2n+31。作为一种特殊情况，不存在不同的序列Γ1，m-长度为λ m+2的集合。这些是pointclass Γ2n+1和Γ1，m的最佳结果。

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 : 2025-04-22 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

Completeness in local positive logic 局部正逻辑中的完备性

IF 0.6 2区数学 Q2 LOGIC

Annals of Pure and Applied Logic

Pub Date : 2025-04-08 DOI: 10.1016/j.apal.2025.103601

Arturo Rodríguez Fanlo , Ori Segel

We develop the basic model theory of local positive logic, a new logic that mixes positive logic (where negation is not allowed) and local logic (where models omit types of infinite distant pairs). We study several basic model theoretic notions such as compactness, positive closedness (existential closedness) and completeness (irreducibility).

我们发展了局部积极逻辑的基本模型理论，这是一种混合了积极逻辑（其中否定是不允许的）和局部逻辑（其中模型省略了无限远对的类型）的新逻辑。研究了紧性、正闭性（存在闭性）和完备性（不可约性）等模型理论的基本概念。

引用次数: 0

The Set-Cover game and non-measurable unions Set-Cover游戏和不可衡量的工会

IF 0.6 2区数学 Q2 LOGIC

Annals of Pure and Applied Logic

Pub Date : 2025-04-07 DOI: 10.1016/j.apal.2025.103602

Taras Banakh , Robert Rałowski , Szymon Żeberski

Using a game-theoretic approach we present a generalization of the classical result of Brzuchowski, Cichoń, Grzegorek and Ryll-Nardzewski on non-measurable unions. We also present applications of obtained results to Marczewski–Burstin representable ideals.

本文利用博弈论的方法推广了Brzuchowski、cichoski、Grzegorek和Ryll-Nardzewski关于不可测并的经典结果。我们还介绍了所得结果在马尔切夫斯基-布尔斯汀可表征理想中的应用。

引用次数: 0

Some more results on relativized Chaitin's Ω 关于相对化柴廷的Ω的更多结果

IF 0.6 2区数学 Q2 LOGIC

Annals of Pure and Applied Logic

Pub Date : 2025-04-04 DOI: 10.1016/j.apal.2025.103586

Liang Yu

We prove that, assuming ZF, and restricted to any

\leq_{T}

-pointed set, Chaitin's

Ω_{U} : x \mapsto Ω_{U}^{x} = \sum_{U^{x} (σ) ↓} 2^{- | σ |}

is not injective for any universal prefix-free Turing machine U, and that

Ω_{U}^{x}

fails to be degree invariant in a very strong sense, answering several recent questions in descriptive set theory. Moreover, we show that under

ZF + AD

, every function f mapping x to x-random must be uncountable-to-one over an upper cone of Turing degrees.

我们证明，假设 ZF，并限制于任何 ≤T 点集，柴廷的ΩU:x↦ΩUx=∑Ux(σ)↓2-|σ| 对于任何通用无前缀图灵机 U 都不是注入式的，并且ΩUx 在非常强的意义上不具有度不变性，这回答了描述集合论中最近的几个问题。此外，我们还证明了在 ZF+AD 下，映射 x 到 x-random 的每个函数 f 都必须在图灵度的上锥上是不可数到一的。

引用次数: 0

Generics in invariant subsets of the group of order preserving permutations of Q Q的保序置换群的不变子集中的泛型

IF 0.6 2区数学 Q2 LOGIC

Annals of Pure and Applied Logic

Pub Date : 2025-03-25 DOI: 10.1016/j.apal.2025.103585

M. Drzewiecka , A. Ivanov , B. Mokry

Let

ρ \in Aut (Q, <)

and

C_{ρ}

be the closure of the conjugacy class of ρ in

Aut (Q, <)

. We show that

C_{ρ}

contains a conjugacy class, say C, which is comeagre in

C_{ρ}

. We describe representatives of C. Furthermore, we show that the family of finite partial maps extendable to elements of C has the cofinal amalgamation property.

设ρ∈Aut(Q,<)，且ρ是ρ在Aut（Q,<）中的共轭类的闭包。我们证明了C包含一个共轭类，比如C，它在C中是相合的。我们描述了C的代表，进一步证明了可扩展到C元素的有限部分映射族具有共终合并性质。

引用次数: 0

Extender-based Magidor-Radin forcings without top extenders 基于扩展器的Magidor-Radin强制装置，无需顶部扩展器

IF 0.6 2区数学 Q2 LOGIC

Annals of Pure and Applied Logic

Pub Date : 2025-03-21 DOI: 10.1016/j.apal.2025.103584

Moti Gitik , Sittinon Jirattikansakul

Continuing [1], we develop a version of Extender-based Magidor-Radin forcing where there are no extenders on the top ordinal. As an application, we provide another approach to obtain a failure of SCH on a club subset of an inaccessible cardinal, and a model where the cardinal arithmetic behaviors are different on stationary classes, whose union is the club, is provided. The cardinals and the cofinalities outside the clubs are not affected by the forcings.

继续[1]，我们开发了一个基于扩展器的Magidor-Radin强制版本，其中顶部序数上没有扩展器。作为应用，我们提供了另一种方法来获得不可达基数的俱乐部子集上的SCH失败，并提供了一个基数算术行为不同的平稳类的模型，这些平稳类的联合是俱乐部。枢机主教和俱乐部外的共谋者不受强迫的影响。

引用次数: 0

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