Annals of Pure and Applied Logic最新文献

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On classification of continuous first order theories 关于连续一阶理论的分类

IF 0.6 2区数学 Q2 LOGIC

Annals of Pure and Applied Logic

Pub Date : 2026-02-02 DOI: 10.1016/j.apal.2026.103729

Karim Khanaki

We present several new characterizations of IP (the independence property) and SOP (the strict order property) for continuous first-order logic and explore their connections to functional analysis and Banach space theory. Furthermore, we propose new dividing lines for unstable theories by examining subclasses of Baire-1 functions. We also explain why one should not expect a perfect analogue of Shelah's theorem—namely, that a theory is unstable if and only if it has IP or SOP—to hold for real-valued logics, particularly in the context of continuous logic.

给出了连续一阶逻辑的独立性和严格序性的几个新特征，并探讨了它们与泛函分析和Banach空间理论的联系。此外，我们通过检验Baire-1函数的子类，提出了不稳定理论的新分界线。我们也解释了为什么我们不应该期待一个完美的类似于Shelah定理——也就是说，一个理论是不稳定的当且仅当它有IP或sop——对于实值逻辑，特别是在连续逻辑的背景下。

引用次数: 0

Positively closed Sh(B)-valued models 正闭Sh(B)值模型

IF 0.6 2区数学 Q2 LOGIC

Annals of Pure and Applied Logic

Pub Date : 2026-01-28 DOI: 10.1016/j.apal.2026.103726

Kristóf Kanalas

We study positively closed and strongly positively closed topos-valued models of coherent theories. Positively closed is a global notion (it is defined in terms of all possible outgoing homomorphisms), while strongly positively closed is a local notion (it only concerns the definable sets inside the model). For Set-valued models of coherent theories they coincide.

We prove that if

E = S h (B, τ_{c o h})

for a complete Boolean algebra, then positively closed but not strongly positively closed

E

-valued models of coherent theories exist, yet, there is an alternative local property which characterizes positively closed

E

-valued models.

A large part of our discussion is given in the context of infinite quantifier geometric logic, dealing with the fragment

L_{κ κ}^{g}

where κ is weakly compact.

我们研究了相干理论的正闭和强正闭拓扑值模型。正闭是一个全局概念（它是根据所有可能的外向同态定义的），而强正闭是一个局部概念（它只涉及模型内的可定义集合）。对于相干理论的集值模型，它们是一致的。我们证明了对于完全布尔代数，如果E=Sh(B,τcoh)，则相干理论的正闭但不是强正闭E值模型存在，然而，存在表征正闭E值模型的另一种局部性质。我们的大部分讨论是在无限量词几何逻辑的背景下给出的，处理片段Lκκg，其中κ是弱紧的。

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引用次数: 0

On algebraic sums, trees and ideals in the Cantor space 关于康托空间中的代数和、树和理想

IF 0.6 2区数学 Q2 LOGIC

Annals of Pure and Applied Logic

Pub Date : 2026-01-12 DOI: 10.1016/j.apal.2026.103724

Marcin Michalski, Robert Rałowski, Szymon Żeberski

We work in the Cantor space

2^{ω}

. The results of the paper adhere to the following pattern. Let

I \in {M, N, M \cap N, E}

and T be a perfect, uniformly perfect or Silver tree. Then for every

A \in I

there exists

T^{'} \subseteq T

of the same kind as T such that

A + \underset{n –times}{\underset{︸}{[T^{'}] + [T^{'}] + \dots + [T^{'}]}} \in I

for each

n \in ω

. We also prove weaker statements for splitting trees. For the case

E

we also provide a simple characterization of a basis of

E

. We use these results to prove that the algebraic sum of a generalized Luzin set and a generalized Sierpiński set belongs to

u_{0}

and

v_{0}

, provided that

c

is a regular cardinal.

我们在康托空间2ω中工作。本文的结果遵循以下模式。设I∈{M，N,M∩N,E}， T是一棵完全、一致完全或银树。则对于每一个A∈I，存在与T同类的T’，使得A+[T’]+[T’]+…+[T’]︸n倍∈I，对于每一个n∈ω。我们也证明了劈树的弱命题。对于E，我们也给出了E的基的一个简单表征。我们用这些结果证明了广义Luzin集和广义Sierpiński集的代数和属于u0和v0，只要c是正则基数。

引用次数: 0

Constructibility real degrees in the side-by-side Sacks model 并行Sacks模型的可构造性实数

IF 0.6 2区数学 Q2 LOGIC

Annals of Pure and Applied Logic

Pub Date : 2026-01-08 DOI: 10.1016/j.apal.2026.103722

Lorenzo Notaro

We study the join-semilattice of constructibility real degrees in the side-by-side Sacks model, i.e., the model of set theory obtained by forcing with a countable-support product of infinitely many Sacks forcings over the constructible universe. In particular, we prove that in the side-by-side Sacks model the join-semilattice of constructibility real degrees is rigid, i.e., it does not have non-trivial automorphisms.

研究了可构造宇宙上无限多个Sacks强迫的可数支撑积的并行Sacks模型中可构造实数度的联合半格。特别地，我们证明了在并行Sacks模型中构造实数的连接半格是刚性的，即它不存在非平凡自同构。

引用次数: 0

Amorphous sets and dual Dedekind finiteness 无定形集与对偶Dedekind有限性

IF 0.6 2区数学 Q2 LOGIC

Annals of Pure and Applied Logic

Pub Date : 2026-01-08 DOI: 10.1016/j.apal.2026.103723

Yifan Hu , Ruihuan Mao , Guozhen Shen

A set A is dually Dedekind finite if every surjection from A onto A is injective; otherwise, A is dually Dedekind infinite. An amorphous set is an infinite set that cannot be partitioned into two infinite subsets. A strictly amorphous set is an amorphous set in which every partition has only finitely many non-singleton blocks. It is proved consistent with

ZF

(i.e., Zermelo–Fraenkel set theory without the axiom of choice) that there exists an amorphous set A whose power set

P (A)

is dually Dedekind infinite, which gives a negative solution to a question proposed by Truss (1974) [13]. Nevertheless, we prove in

ZF

that, for all strictly amorphous sets A and all natural numbers n,

P {(A)}^{n}

is dually Dedekind finite, which generalizes a result of Goldstern.

如果从A到A的所有射都是内射，则集合A是对偶Dedekind有限的；否则，A是对偶Dedekind无穷大。无定形集是一种不能被分割成两个无限子集的无限集。严格无定形集是一种无定形集，其中每个划分只有有限多个非单块。证明了存在一个幂集P(A)为对偶Dedekind无穷的无定形集A，与ZF（即不带选择公理的Zermelo-Fraenkel集合理论）是一致的，给出了Truss(1974)[13]问题的一个负解。然而，我们在ZF中证明了，对于所有严格无定形集合A和所有自然数n， P(A)n是对偶Dedekind有限，推广了Goldstern的结果。

引用次数: 0

The category dichotomy for ideals 理想的范畴二分法

IF 0.6 2区数学 Q2 LOGIC

Annals of Pure and Applied Logic

Pub Date : 2026-01-07 DOI: 10.1016/j.apal.2025.103717

Alan Dow , Raúl Figueroa-Sierra , Osvaldo Guzmán , Michael Hrušák

We prove that there is an ideal on ω that is not Katětov below nwd and does not have restrictions above

ED

. We also prove that in the Laver model every tall P-ideal is Katětov-Blass above

{ED}_{fin}

and that it is consistent that every Q⁺ ideal is meager.

我们证明了ω上存在一个理想，在nwd以下不为katkattov，在ED以上不受限制。我们还证明了Laver模型中每个高p理想在EDfin以上都是katkattov - blass，并且每个Q+理想都是贫乏的是一致的。

引用次数: 0

Preserving and constructing multiple gaps 保留和建造多个缺口

IF 0.6 2区数学 Q2 LOGIC

Annals of Pure and Applied Logic

Pub Date : 2026-01-07 DOI: 10.1016/j.apal.2026.103721

Osvaldo Guzmán , Francisco Santiago Nieto-de la Rosa

We analyze conditions to preserve with forcing multiple gaps, define conditions to ensure their existence, and study the minimal size of specific type of gaps.

分析了强制多间隙的保存条件，定义了保证多间隙存在的条件，研究了特定类型间隙的最小尺寸。

引用次数: 0

Algebraic independence of the solutions of the classical Lotka-Volterra system 经典Lotka-Volterra系统解的代数独立性

IF 0.6 2区数学 Q2 LOGIC

Annals of Pure and Applied Logic

Pub Date : 2026-01-02 DOI: 10.1016/j.apal.2025.103707

Yutong Duan, Joel Nagloo

Let

(x_{1}, y_{1}), \dots, (x_{n}, y_{n})

be distinct non-constant and non-degenerate solutions of the classical Lotka-Volterra system

x^{'} = a x y + b x y^{'} = c x y + d y

where

a, b, c, d \in C ∖ {0}

. We show that if d and b are linearly independent over

Q

, then the solutions are algebraically independent over

C

, that is

t r . d e g_{C} C (x_{1}, y_{1}, \dots, x_{n}, y_{n}) = 2 n

. As a main part of the proof, we show that the set defined by the system in universal differential fields, with d and b linearly independent over

Q

, is strongly minimal and geometrically trivial. Our techniques also allows us to obtain partial results for some of the more general 2d-Lotka-Volterra system.

设（x1,y1），…，（xn,yn）是经典Lotka-Volterra方程组x ' =axy+bxy ' =cxy+dy的不同非常非退化解，其中a，b,c,d∈c∈{0}。我们证明，如果d和b在Q上是线性无关的，那么它们的解在C上是代数无关的，即trd . degcc (x1,y1，…，xn,yn)=2n。作为证明的主要部分，我们证明了在泛微分域中，当d和b在Q上线性无关时，系统所定义的集合是强极小和几何平凡的。我们的技术也允许我们获得一些更一般的2d-Lotka-Volterra系统的部分结果。

引用次数: 0

Constructive quantifier elimination with a focus on matrix rings 以矩阵环为重点的建设性量词消去

IF 0.6 2区数学 Q2 LOGIC

Annals of Pure and Applied Logic

Pub Date : 2025-12-22 DOI: 10.1016/j.apal.2025.103706

Max Illmer, Tim Netzer

We give a sufficient condition for a model theoretic structure B to ‘inherit’ quantifier elimination from another structure A. This yields an alternative proof of one of the main results from [8], namely quantifier elimination for certain matrix rings. The original proof uses model theory, and while it is very elegant and insightful, the proof we propose is much shorter and provides a constructive algorithm.

我们给出了模型理论结构B从另一个结构a“继承”量词消去的一个充分条件。这产生了[8]的一个主要结果的替代证明，即某些矩阵环的量词消去。最初的证明使用了模型理论，虽然它非常优雅和富有洞察力，但我们提出的证明要短得多，并提供了一个建设性的算法。

引用次数: 0

A note on Erdős-Hajnal property for graphs with VC dimension ≤2 关于VC维≤2图Erdős-Hajnal性质的注解

IF 0.6 2区数学 Q2 LOGIC

Annals of Pure and Applied Logic

Pub Date : 2025-12-16 DOI: 10.1016/j.apal.2025.103705

Yayi Fu

Using a result in [6] and the proof of [2, Theorem 1.1], we show that there is

γ > 0

such that for any finite graph G with VC-dimension ≤2, G has a clique or an anti-clique of size

\geq | G |^{γ}

. We also show that the Erdős-Hajnal property for graphs with VC-dimension 1 can be proved using the δ-dimension technique in [4], and we show that when E is a definable symmetric binary relation, [4, Theorem 1.3] can be proved without using Shelah's 2-rank.

利用[6]中的一个结果和[2，定理1.1]的证明，我们证明了γ>；0使得对于任何vc维≤2的有限图G， G存在一个大小≥|G|γ的团或反团。我们还证明了在[4]中用δ维技术证明了vc维数为1的图的Erdős-Hajnal性质，并且证明了当E是一个可定义的对称二元关系时，[4，定理1.3]可以不使用Shelah的2-秩证明。

引用次数: 0

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