Annals of Pure and Applied Logic最新文献_第8页

Classification of ℵ0-categorical C-minimal pure C-sets c -极小纯c集的分类

IF 0.8 2区数学 Q2 Mathematics

Annals of Pure and Applied Logic

Pub Date : 2023-09-27 DOI: 10.1016/j.apal.2023.103375

Françoise Delon , Marie-Hélène Mourgues

We classify all $ℵ_{0}$ -categorical and C-minimal C-sets up to elementary equivalence. As usual the Ryll-Nardzewski Theorem makes the classification of indiscernible $ℵ_{0}$ -categorical C-minimal sets as a first step. We first define solvable good trees, via a finite induction. The trees involved in initial and induction steps have a set of nodes, either consisting of a singleton, or having dense branches without endpoints and the same number of branches at each node. The class of colored good trees is the elementary class of solvable good trees. We show that a pure C-set M is indiscernible, finite or $ℵ_{0}$ -categorical and C-minimal iff its canonical tree $T (M)$ is a colored good tree. The classification of general $ℵ_{0}$ -categorical and C-minimal C-sets is done via finite trees with labeled vertices and edges, where labels are natural numbers, or infinity and complete theories of indiscernible, $ℵ_{0}$ -categorical or finite, and C-minimal C-sets.

我们对所有ℵ0-范畴和C-极小C-集直至初等等价。像往常一样，Ryll-Nardzewski定理使得不可分辨的分类ℵ0-范畴C-极小集作为第一步。我们首先通过有限归纳定义了可解的好树。初始步骤和归纳步骤中涉及的树有一组节点，要么由单个节点组成，要么具有没有端点的密集分支，每个节点的分支数量相同。有色好树类是可解好树的初等类。我们证明了纯C集M是不可分辨的、有限的或ℵ0-范畴和C-极小当其正则树T（M）是有色好树。一般的分类ℵ0-范畴和C-极小C-集是通过具有标记顶点和边的有限树来实现的，其中标记是自然数，或无穷大和不可分辨的完全理论，ℵ0-范畴或有限的C-极小C-集。

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引用次数: 0

Positive modal logic beyond distributivity 超越分布性的正模态逻辑

IF 0.8 2区数学 Q2 Mathematics

Annals of Pure and Applied Logic

Pub Date : 2023-09-26 DOI: 10.1016/j.apal.2023.103374

Nick Bezhanishvili , Anna Dmitrieva , Jim de Groot , Tommaso Moraschini

We develop a duality for (modal) lattices that need not be distributive, and use it to study positive (modal) logic beyond distributivity, which we call weak positive (modal) logic. This duality builds on the Hofmann, Mislove and Stralka duality for meet-semilattices. We introduce the notion of $Π_{1}$ -persistence and show that every weak positive modal logic is $Π_{1}$ -persistent. This approach leads to a new relational semantics for weak positive modal logic, for which we prove an analogue of Sahlqvist's correspondence result.¹

我们为不需要分配的（模态）格发展了对偶性，并用它来研究超越分配性的正（模态）逻辑，我们称之为弱正（模式）逻辑。这种对偶建立在满足半格的Hofmann、Mislove和Stralka对偶的基础上。我们引入了π1-持久性的概念，并证明了每一个弱正模态逻辑都是π1-持久的。这种方法为弱正模态逻辑带来了一种新的关系语义，我们证明了其类似于Sahlqvist的对应结果。1

引用次数: 4

Weak saturation properties and side conditions 弱饱和特性和侧面条件

IF 0.8 2区数学 Q2 Mathematics

Annals of Pure and Applied Logic

Pub Date : 2023-09-06 DOI: 10.1016/j.apal.2023.103356

Monroe Eskew

Towards combining “compactness” and “hugeness” properties at $ω_{2}$ , we investigate the relevance of side-conditions forcing. We reduce the upper bound on the consistency strength of the weak Chang's Conjecture at $ω_{2}$ using Neeman's forcing. On the other hand, we find a barrier to the applicability of these methods to our problem and give a counterexample to a claim of Neeman about the effects of iterating such forcing.

为了结合ω2的“紧性”和“巨大性”性质，我们研究了侧条件强迫的相关性。我们利用尼曼力降低了弱张氏猜想在ω2处的一致性强度的上界。另一方面，我们发现了将这些方法应用于我们的问题的一个障碍，并给出了尼曼关于迭代这种强迫的影响的主张的反例。

引用次数: 0

Absoluteness for the theory of the inner model constructed from finitely many cofinality quantifiers 由有限多个共定性量词构造的内部模型理论的绝对性

IF 0.8 2区数学 Q2 Mathematics

Annals of Pure and Applied Logic

Pub Date : 2023-09-01 DOI: 10.1016/j.apal.2023.103358

Ur Ya'ar

We prove that the theory of the models constructible using finitely many cofinality quantifiers – $C_{λ_{1}, \dots, λ_{n}}^{⁎}$ and $C_{< λ_{1}, \dots, < λ_{n}}^{⁎}$ for $λ_{1}, \dots, λ_{n}$ regular cardinals – is set-forcing absolute under the assumption of class many Woodin cardinals, and is independent of the regular cardinals used. Towards this goal we prove some properties of the generic embedding induced from the stationary tower restricted to <μ-closed sets.

证明了λ1，…，λn和λ1，…，λn对λ1，…，λn的有限多个共度量词模型的理论在类多Woodin基数的假设下是集强迫绝对的，并且与所使用的正则基数无关。为此，我们证明了由μ闭集约束的固定塔导出的一般嵌入的一些性质。

引用次数: 0

Generalized fusible numbers and their ordinals 广义可熔数及其序数

IF 0.8 2区数学 Q2 Mathematics

Annals of Pure and Applied Logic

Pub Date : 2023-09-01 DOI: 10.1016/j.apal.2023.103355

Alexander I. Bufetov , Gabriel Nivasch , Fedor Pakhomov

Erickson defined the fusible numbers as a set $F$ of reals generated by repeated application of the function $\frac{x + y + 1}{2}$ . Erickson, Nivasch, and Xu showed that $F$ is well ordered, with order type $ε_{0}$ . They also investigated a recursively defined function $M : R \to R$ . They showed that the set of points of discontinuity of M is a subset of $F$ of order type $ε_{0}$ . They also showed that, although M is a total function on $R$ , the fact that the restriction of M to $Q$ is total is not provable in first-order Peano arithmetic $PA$ .

In this paper we explore the problem (raised by Friedman) of whether similar approaches can yield well-ordered sets $F$ of larger order types. As Friedman pointed out, Kruskal's tree theorem yields an upper bound of the small Veblen ordinal for the order type of any set generated in a similar way by repeated application of a monotone function $g : R^{n} \to R$ .

The most straightforward generalization of $\frac{x + y + 1}{2}$ to an n-ary function is the function $\frac{x_{1} + \dots + x_{n} + 1}{n}$ . We show that this function generates a set $F_{n}$ whose order type is just $φ_{n - 1} (0)$ . For this, we develop recursively defined functions $M_{n} : R \to R$ naturally generalizing the function M.

Furthermore, we prove that for any linear function $g : R^{n} \to R$ , the order type of the resulting $F$ is at most $φ_{n - 1} (0 <$

Erickson将可熔数定义为通过重复应用函数x+y+12生成的实数集F。Erickson、Nivasch和Xu证明了F是有序的，其有序类型为ε0。他们还研究了一个递归定义的函数M:R→R。他们证明了M的不连续点集是阶型ε0的F的子集。他们还证明，尽管M是R上的一个全函数，但在一阶Peano算术PA中，M对Q的限制是全的这一事实是不可证明的。在本文中，我们探讨了类似方法是否可以产生更高阶类型的良序集F的问题（由Friedman提出）。正如Friedman所指出的，Kruskal树定理为通过重复应用单调函数g:Rn以类似方式生成的任何集合的阶类型产生了小Veblen序数的上界→R.x+y+12对n元函数最直接的推广是函数x1+…+xn+1n。我们证明了这个函数生成了一个集合Fn，它的阶型恰好是φn-1（0）。为此，我们开发了递归定义的函数Mn:R→R自然地推广了函数M。此外，我们证明了对于任何线性函数g:Rn→R、得到的F的阶型至多为φn−1（0）。最后，我们证明了确实存在连续函数g:Rn→R，其结果集F的阶类型接近小的Veblen序数。

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引用次数: 0

Extensions of Solovay's system S without independent sets of axioms 没有独立公理集的Solovay系统s的扩张

IF 0.8 2区数学 Q2 Mathematics

Annals of Pure and Applied Logic

Pub Date : 2023-09-01 DOI: 10.1016/j.apal.2023.103360

Igor Gorbunov , Dmitry Shkatov

Chagrov and Zakharyaschev posed the problem of existence of extensions of Solovay's system S, which is a non-normalizable quasi-normal modal logic, that do not admit deductively independent sets of axioms. This paper gives a solution by exhibiting countably many extensions of S without deductively independent sets of axioms.

Chagrov和Zakharyaschev提出了Solovay系统S的扩展的存在性问题，Solovay系统S是一个不可归一化的拟正态模态逻辑，它不允许演绎独立的公理集。本文给出了一个解，给出了无演绎独立公理集的S的可数扩展。

引用次数: 0

Indestructibility of some compactness principles over models of PFA PFA模型上一些紧致原则的不可破坏性

IF 0.8 2区数学 Q2 Mathematics

Annals of Pure and Applied Logic

Pub Date : 2023-08-30 DOI: 10.1016/j.apal.2023.103359

Radek Honzik , Chris Lambie-Hanson , Šárka Stejskalová

We show that

PFA

(Proper Forcing Axiom) implies that adding any number of Cohen subsets of ω will not add an

ω_{2}

-Aronszajn tree or a weak

ω_{1}

-Kurepa tree, and moreover no σ-centered forcing can add a weak

ω_{1}

-Kurepa tree (a tree of height and size

ω_{1}

with at least

ω_{2}

cofinal branches). This partially answers an open problem whether ccc forcings can add

ω_{2}

-Aronszajn trees or

ω_{1}

-Kurepa trees (with

\neg □_{ω_{1}}

in the latter case).

We actually prove more: We show that a consequence of $PFA$ , namely the guessing model principle, $GMP$ , which is equivalent to the ineffable slender tree property, $ISP$ , is preserved by adding any number of Cohen subsets of ω. And moreover, $GMP$ implies that no σ-centered forcing can add a weak $ω_{1}$ -Kurepa tree (see Section 2.1 for definitions).

For more generality, we study variations of the principle $GMP$ at higher cardinals and the indestructibility consequences they entail, and as applications we answer a question of Mohammadpour about guessing models at weakly but not strongly inaccessible cardinals and show that there is a model in which there are no weak $ℵ_{ω + 1}$ -Kurepa trees and no $ℵ_{ω + 2}$ -Aronszajn trees.

我们证明了PFA（Proper Forcing Axiom）意味着添加任意数量的ω的Cohen子集不会添加ω2-Aronszajn树或弱ω1-Kurepa树，而且没有σ-中心强迫可以添加弱ω1-Kurepa树（高度和大小为ω1且至少有ω2共尾分支的树）。这部分回答了一个悬而未决的问题，即ccc强迫是否可以添加ω2-Aronszajn树或ω1-Kurepa树（□在后一种情况下为ω1）。我们实际上证明了更多：我们证明了PFA的一个结果，即猜测模型原理GMP，它等价于无法形容的细长树性质ISP，通过添加ω的任意数量的Cohen子集来保持。此外，GMP意味着没有以σ为中心的强迫可以添加弱ω1-Kurepa树（定义见第2.1节）。为了更普遍，我们研究了原则GMP在更高基数下的变化及其带来的不可破坏性后果，作为应用，我们回答了Mohammadpour关于在弱但非强不可访问基数上猜测模型的问题，并证明了存在一个不存在弱ℵω+1-Kurepa树和noℵω+2-Aronszajn树。

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引用次数: 0

On countably perfectly meager and countably perfectly null sets 关于可数完全贫乏集和可数完全空集

IF 0.8 2区数学 Q2 Mathematics

Annals of Pure and Applied Logic

Pub Date : 2023-08-29 DOI: 10.1016/j.apal.2023.103357

Tomasz Weiss , Piotr Zakrzewski

We study a strengthening of the notion of a universally meager set and its dual counterpart that strengthens the notion of a universally null set.

We say that a subset A of a perfect Polish space X is countably perfectly meager (respectively, countably perfectly null) in X, if for every perfect Polish topology τ on X, giving the original Borel structure of X, A is covered by an $F_{σ}$ -set F in X with the original Polish topology such that F is meager with respect to τ (respectively, for every finite, non-atomic, Borel measure μ on X, A is covered by an $F_{σ}$ -set F in X with $μ (F) = 0$ ).

We prove that if $2^{ℵ_{0}} \leq ℵ_{2}$ , then there exists a universally meager set in $2^{N}$ which is not countably perfectly meager in $2^{N}$ (respectively, a universally null set in $2^{N}$ which is not countably perfectly null in $2^{N}$ ).

我们研究了对普遍贫乏集概念的强化及其对偶对偶强化了普遍零集的概念。我们说一个完美的波兰空间X是一个子集可数完美的(可数完美零)分别在X,如果每一个完美的波兰拓扑τX, X的原始波莱尔结构,是由一个Fσ集F在X与原波兰拓扑,F是微薄对τ(分别为每一个有限的、非原子波莱尔测量μX,覆盖着一个Fσ组XμF (F) = 0)。证明了如果2≤2，则在2N中存在一个在2N中不可数完全贫乏的普遍贫乏集(即在2N中存在一个在2N中不可数完全贫乏的普遍零集)。

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引用次数: 0

Primitive recursive reverse mathematics 原始递归逆数学

IF 0.8 2区数学 Q2 Mathematics

Annals of Pure and Applied Logic

Pub Date : 2023-08-25 DOI: 10.1016/j.apal.2023.103354

Nikolay Bazhenov , Marta Fiori-Carones , Lu Liu , Alexander Melnikov

We use a second-order analogy ${PRA}^{2}$ of $PRA$ to investigate the proof-theoretic strength of theorems in countable algebra, analysis, and infinite combinatorics. We compare our results with similar results in the fast-developing field of primitive recursive (‘punctual’) algebra and analysis, and with results from ‘online’ combinatorics. We argue that ${PRA}^{2}$ is sufficiently robust to serve as an alternative base system below ${RCA}_{0}$ to study the proof-theoretic content of theorems in ordinary mathematics. (The most popular alternative is perhaps ${RCA}_{0}^{⁎}$ .) We discover that many theorems that are known to be true in ${RCA}_{0}$ either hold in ${PRA}^{2}$ or are equivalent to ${RCA}_{0}$ or its weaker (but natural) analogy $2^{N}$ - ${RCA}_{0}$ over ${PRA}^{2}$ . However, we also discover that some standard mathematical and combinatorial facts are incomparable with these natural subsystems.

我们使用PRA的二阶类比PRA2来研究可数代数、分析和无限组合中定理的证明理论强度。我们将我们的结果与快速发展的原始递归(“准时”)代数和分析领域的类似结果以及“在线”组合学的结果进行比较。我们认为PRA2具有足够的鲁棒性，可以作为RCA0之下的备选基系统来研究普通数学中定理的证明理论内容。(最流行的替代方案可能是RCA0。)我们发现许多在RCA0中已知为真的定理在PRA2中也成立，或者等价于RCA0或其较弱的(但自然的)类比2N-RCA0优于PRA2。然而，我们也发现一些标准的数学和组合事实与这些自然的子系统是无法比较的。

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引用次数: 0

Large cardinals at the brink 大型红雀濒临灭绝

IF 0.8 2区数学 Q2 Mathematics

Annals of Pure and Applied Logic

Pub Date : 2023-08-10 DOI: 10.1016/j.apal.2023.103328

W. Hugh Woodin

Kunen's theorem that assuming the Axiom of Choice there are no Reinhardt cardinals is a key milestone in the program to understand large cardinal axioms. But this theorem is not the end of a story, rather it is the beginning.

库宁定理假设选择公理不存在莱因哈特基数，这是理解大基数公理的一个重要里程碑。但这个定理并不是故事的结尾，而是一个开始。

引用次数: 0

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