Annals of Pure and Applied Logic最新文献

英文中文

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 : 2025-08-01 Epub 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

P-measures in models without P-points 没有p点的模型中的p测度

IF 0.6 2区数学 Q2 LOGIC

Annals of Pure and Applied Logic

Pub Date : 2025-07-01 Epub Date: 2025-03-10 DOI: 10.1016/j.apal.2025.103579

Piotr Borodulin-Nadzieja , Jonathan Cancino-Manríquez , Adam Morawski

We answer in negative the problem if the existence of a P-measure implies the existence of a P-point. Namely, we show that if we add random reals to a certain ‘unique P-point’ model, then in the resulting model we will have a P-measure but not P-points. Also, we investigate the question if there is a P-measure in the Silver model. We show that rapid filters cannot be extended to a P-measure in the extension by ω product of Silver forcings and that in the model obtained by the countable support

ω_{2}

-iteration of countable product of Silver forcings there are no P-measures of countable Maharam type.

如果p测度的存在意味着p点的存在，我们就否定地回答这个问题。也就是说，我们表明，如果我们将随机实数添加到某个“唯一p点”模型中，那么在得到的模型中，我们将有一个p测度，但没有p点。此外，我们还研究了在Silver模型中是否存在p测度的问题。我们证明了快速滤波器在由Silver强迫的ω积进行扩展时不能推广到p测度，并且在由Silver强迫的可数积的ω2次迭代得到的模型中不存在可数Maharam型的p测度。

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 : 2025-07-01 Epub 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

A complete invariant system for noetherian BL-algebras and more general L-algebras noether bl -代数和更一般的l -代数的完全不变系统

IF 0.6 2区数学 Q2 LOGIC

Annals of Pure and Applied Logic

Pub Date : 2025-07-01 Epub Date: 2025-03-07 DOI: 10.1016/j.apal.2025.103580

Wolfgang Rump

Main results on BL-algebras, including their classification in the finite case, are reconsidered and extended to a class of L-algebras X with prime factorization, including BL-algebras with ascending chain condition for its lattice. The weighted forest associated with a finite BL-algebra reappears as a canonical L-subalgebra

\tilde{P} (X)

of prime elements in the self-similar closure

S (X)

where

\tilde{P} (X)

is completely determined by its underlying poset (not necessarily a forest), while the weights are associated with existing powers of the prime elements in X. These invariants determine X within its self-similar closure

S (X) = S (\tilde{P} (X))

. The three basic types of BL-algebras are related to concepts of L-algebras with further-reaching significance in quantum theory.

重新考虑了关于bl -代数的主要结果，包括它们在有限情况下的分类，并将其推广到一类具有素分解的l -代数X，包括其格具有升链条件的bl -代数。与有限bl代数相关的加权森林在自相似闭包S(X)中表现为素数元素的正则l子代数P ~ (X)，其中P ~ (X)完全由其基础偏序集（不一定是森林）决定，而权重与X中素数元素的现有幂相关。这些不变量决定了X在其自相似闭包S(X)=S（P ~ (X)）内。l -代数的三种基本类型与l -代数的概念有关，在量子理论中具有深远的意义。

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引用次数: 0

Cardinal characteristics on bounded generalised Baire spaces 有界广义Baire空间上的基数特征

IF 0.6 2区数学 Q2 LOGIC

Annals of Pure and Applied Logic

Pub Date : 2025-07-01 Epub Date: 2025-03-12 DOI: 10.1016/j.apal.2025.103582

Tristan van der Vlugt

We will give an overview of four families of cardinal characteristics defined on subspaces

\prod_{α \in κ} b (α)

of the generalised Baire space

{}^{κ}κ

, where κ is strongly inaccessible and

b \in^{κ} κ

. The considered families are bounded versions of the dominating, eventual difference, localisation and antilocalisation numbers, and their dual cardinals. We investigate parameters for which these cardinals are nontrivial and how the cardinals relate to each other and to other cardinals of the generalised Cichoń diagram. Finally we prove that different choices of parameters may lead to consistently distinct cardinals.

我们将概述在广义Baire空间κκ的子空间∏α∈κb（α）上定义的四个基本特征族，其中κ是强不可达的，b∈κκ。所考虑的族是主导、最终差异、本地化和反本地化数及其双基数的有限版本。我们研究了这些基数是非平凡的参数，以及这些基数彼此之间以及与广义cichoski图的其他基数之间的关系。最后，我们证明了不同的参数选择可能导致一致的不同基数。

引用次数: 0

Iterated reduced powers of collapsing algebras 塌缩代数的迭代约简幂

IF 0.6 2区数学 Q2 LOGIC

Annals of Pure and Applied Logic

Pub Date : 2025-06-01 Epub Date: 2025-02-28 DOI: 10.1016/j.apal.2025.103567

Miloš S. Kurilić

<div><div><math><mrow><mi>rp</mi></mrow><mo>(</mo><mi>B</mi><mo>)</mo></math> denotes the reduced power <math><msup><mrow><mi>B</mi></mrow><mrow><mi>ω</mi></mrow></msup><mo>/</mo><mi>Φ</mi></math> of a Boolean algebra <math><mi>B</mi></math>, where Φ is the Fréchet filter on ω. We investigate iterated reduced powers (<math><msup><mrow><mi>rp</mi></mrow><mrow><mn>0</mn></mrow></msup><mo>(</mo><mi>B</mi><mo>)</mo><mo>=</mo><mi>B</mi></math> and <math><msup><mrow><mi>rp</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>(</mo><mi>B</mi><mo>)</mo><mo>=</mo><mrow><mi>rp</mi></mrow><mo>(</mo><msup><mrow><mi>rp</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>(</mo><mi>B</mi><mo>)</mo><mo>)</mo></math>) of collapsing algebras and our main intention is to classify the algebras <math><msup><mrow><mi>rp</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>(</mo><mrow><mi>Col</mi></mrow><mo>(</mo><mi>λ</mi><mo>,</mo><mi>κ</mi><mo>)</mo><mo>)</mo></math>, <math><mi>n</mi><mo>≥</mo><mn>1</mn></math>, up to isomorphism of their Boolean completions. In particular, assuming that SCH and <math><mi>h</mi><mo>=</mo><msub><mrow><mi>ω</mi></mrow><mrow><mn>1</mn></mrow></msub></math> hold, we show that for any cardinals <math><mi>λ</mi><mo>≥</mo><mi>ω</mi></math> and <math><mi>κ</mi><mo>≥</mo><mn>2</mn></math> such that <math><mi>κ</mi><mi>λ</mi><mo>></mo><mi>ω</mi></math> and <math><mrow><mi>cf</mi></mrow><mo>(</mo><mi>λ</mi><mo>)</mo><mo>≤</mo><mi>c</mi></math> we have <math><mrow><mi>ro</mi></mrow><mo>(</mo><msup><mrow><mi>rp</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>(</mo><mrow><mi>Col</mi></mrow><mo>(</mo><mi>λ</mi><mo>,</mo><mi>κ</mi><mo>)</mo><mo>)</mo><mo>)</mo><mo>≅</mo><mrow><mi>Col</mi></mrow><mo>(</mo><msub><mrow><mi>ω</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msup><mrow><mo>(</mo><msup><mrow><mi>κ</mi></mrow><mrow><mo><</mo><mi>λ</mi></mrow></msup><mo>)</mo></mrow><mrow><mi>ω</mi></mrow></msup><mo>)</mo></math>, for each <math><mi>n</mi><mo>≥</mo><mn>1</mn></math>; more precisely,<math><mrow><mi>ro</mi></mrow><mo>(</mo><msup><mrow><mi>rp</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>(</mo><mrow><mi>Col</mi></mrow><mo>(</mo><mi>λ</mi><mo>,</mo><mi>κ</mi><mo>)</mo><mo>)</mo><mo>)</mo><mo>≅</mo><mrow><mo>{</mo><mtable><mtr><mtd><mrow><mi>Col</mi></mrow><mo>(</mo><msub><mrow><mi>ω</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mi>c</mi><mo>)</mo><mo>,</mo><mspace></mspace></mtd><mtd><mtext> if </mtext><msup><mrow><mi>κ</mi></mrow><mrow><mo><</mo><mi>λ</mi></mrow></msup><mo>≤</mo><mi>c</mi><mo>;</mo></mtd></mtr><mtr><mtd><mrow><mi>Col</mi></mrow><mo>(</mo><msub><mrow><mi>ω</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msup><mrow><mi>κ</mi></mrow><mrow><mo><</mo><mi>λ</mi></mrow></msup><mo>)</mo><mo>,</mo><mspace></mspa

rp(B)表示布尔代数B的约简幂Bω/Φ，其中Φ是ω上的fr切特滤波器。我们研究了坍缩代数的迭代约简幂(r0 (B)=B和rpn+1(B)=rp(rpn(B)))，我们的主要目的是对rpn(Col(λ,κ)), n≥1，直至其布尔补全的同构的代数进行分类。特别地，假设SCH和h=ω1成立，我们证明对于任意基数λ≥ω和κ≥2，使得κλ>；ω和cf(λ)≤c，我们有ro(rpn(Col(λ,κ))) (Col(λ,κ))) (Col(ω1,(κ<λ)ω)，对于每个n≥1；更准确地说,ro (rpn (Col(κλ)))≅{坳(ω1 c),如果κ& lt;λ≤c;坳(ω1,κ& lt;λ),如果κ& lt;λ在c∧cf(κ& lt;λ)在ω;坳(ω1,(κ& lt;λ)+),如果κ& lt;λ在c∧cf(κ& lt;λ)=ω。若b=d且0 #不存在，则无论cf(λ)=ω，均成立。

引用次数: 0

Tight cofinitary groups 紧共群

IF 0.6 2区数学 Q2 LOGIC

Annals of Pure and Applied Logic

Pub Date : 2025-06-01 Epub Date: 2025-03-04 DOI: 10.1016/j.apal.2025.103570

Vera Fischer , L. Schembecker , David Schrittesser

We introduce the notion of a tight cofinitary group, which captures forcing indestructibility of maximal cofinitary groups for a long list of partial orders, including Cohen, Sacks, Miller, Miller partition forcing and Shelah's poset for diagonalizing maximal ideals. Introducing a new robust coding technique, we establish the relative consistency of

a_{g} = d < c = ℵ_{2}

alongside the existence of a

Δ_{3}^{1}

-well-order of the reals and a co-analytic witness for

a_{g}

我们引入了紧共有限群的概念，它描述了一长串偏序的极大共有限群的强制不灭性，包括Cohen， Sacks, Miller， Miller分割强迫和最大理想对角化的Shelah偏集。引入一种新的鲁棒编码技术，我们建立了ag=d<c= λ 2的相对一致性，以及ag的一个Δ31-well-order实数和一个协解析见证的存在。

引用次数: 0

Borel sets without perfectly many overlapping translations, III Borel集没有完美的许多重叠翻译，III

IF 0.6 2区数学 Q2 LOGIC

Annals of Pure and Applied Logic

Pub Date : 2025-06-01 Epub Date: 2025-02-21 DOI: 10.1016/j.apal.2025.103565

Andrzej Rosłanowski , Saharon Shelah

We expand the results of Rosłanowski and Shelah [11], [10] to all perfect Abelian Polish groups

(H, +)

. In particular, we show that if

α < ω_{1}

and

4 \leq k < ω

, then there is a ccc forcing notion adding a

Σ_{2}^{0}

set

B \subseteq H

which has

ℵ_{α}

many pairwise k–overlapping translations but not a perfect set of such translations. The technicalities of the forcing construction led us to investigations of the question when, in an Abelian group,

X - X \subseteq Y - Y

imply that a translation of X or −X is included in Y.

我们将Rosłanowski和Shelah[11]，[10]的结果扩展到所有完美的Abelian Polish群（H,+）。特别地，我们证明了如果α<；ω1和4≤k<；ω，则存在一个ccc强迫概念，添加一个Σ20集合B∈H，该集合有多个k对重叠的翻译，但不是一个完美的翻译集。强迫构造的技术性使我们对下述问题进行了研究：在阿贝尔群中，X−X对子Y−Y意味着X或- X的翻译包含在Y中。

引用次数: 0

Universally Sacks-indestructible combinatorial families of reals 普遍萨克斯-不可摧毁的实数组合族

IF 0.6 2区数学 Q2 LOGIC

Annals of Pure and Applied Logic

Pub Date : 2025-06-01 Epub Date: 2025-03-03 DOI: 10.1016/j.apal.2025.103566

V. Fischer , L. Schembecker

We introduce the notion of an arithmetical type of combinatorial family of reals, which serves to generalize different types of families such as mad families, maximal cofinitary groups, ultrafilter bases, splitting families and other similar types of families commonly studied in combinatorial set theory.

We then prove that every combinatorial family of reals of arithmetical type which is indestructible by the product of Sacks forcing

S^{ℵ_{0}}

is in fact universally Sacks-indestructible, i.e. it is indestructible by any countably supported iteration or product of Sacks-forcing of any length. Further, under

CH

we present a unified construction of universally Sacks-indestructible families for various arithmetical types of families. In particular we prove the existence of a universally Sacks-indestructible maximal cofinitary group under

CH

本文引入了实数组合族的算术类型的概念，用于推广组合集理论中常见的疯狂族、极大共有限群、超滤基、分裂族以及其他类似类型的族。然后，我们证明了每一个算术型实数组合族，如果它是由Sacks强迫S的乘积不能被破坏的，那么它实际上是普遍的Sacks-不可破坏的，即它是由任何长度的Sacks强迫的可数支持迭代或乘积不能被破坏的。进一步，在CH条件下，我们给出了各种算术类型族的普遍sacks -不可灭族的统一构造。特别地，我们证明了CH下一个普遍的sacks -不可破极大共群的存在性。

引用次数: 0

A finitary Kronecker's lemma and large deviations in the strong law of large numbers on Banach spaces Banach空间上的有限Kronecker引理和强数定律中的大偏差

IF 0.6 2区数学 Q2 LOGIC

Annals of Pure and Applied Logic

Pub Date : 2025-06-01 Epub Date: 2025-03-04 DOI: 10.1016/j.apal.2025.103569

Morenikeji Neri

We explore the computational content of Kronecker's lemma via the proof-theoretic perspective of proof mining and utilise the resulting finitary variant of this fundamental result to provide new rates for the Strong Law of Large Numbers for random variables taking values in type p Banach spaces, which in particular are very uniform in the sense that they do not depend on the distribution of the random variables. Furthermore, we provide computability-theoretic arguments to demonstrate the ineffectiveness of Kronecker's lemma and investigate the result from the perspective of Reverse Mathematics. In addition, we demonstrate how this ineffectiveness from Kronecker's lemma trickles down to the Strong Law of Large Numbers by providing a construction that shows that computable rates of convergence are not always possible. Lastly, we demonstrate how Kronecker's lemma falls under a class of deterministic formulas whose solution to their Dialectica interpretation satisfies a continuity property and how, for such formulas, one obtains an upgrade principle that allows one to lift computational interpretations of deterministic results to quantitative results for their probabilistic analogue. This result generalises the previous work of the author and Pischke.

我们通过证明挖掘的证明论视角探索克罗内克两难的计算内容，并利用这一基本结果的有限变体，为在 p 型巴拿赫空间取值的随机变量的强大数定律提供新的速率，特别是在不依赖于随机变量分布的意义上，这种速率是非常均匀的。此外，我们还提供了可计算性理论论据来证明克罗内克∞的无效性，并从逆数学的角度研究了这一结果。此外，我们还提供了一个构造，说明可计算的收敛率并不总是可能的，以此证明克罗内克两难的无效性是如何向下渗透到大数强律的。最后，我们证明了克罗内克两难如何属于一类确定性公式，其辩证解释的解满足连续性属性，以及对于这类公式，我们如何获得一个升级原理，允许我们将确定性结果的计算解释提升为其概率类似的定量结果。这一结果概括了作者和皮施克之前的工作。

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Annals of Pure and Applied Logic

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