The Journal of Symbolic Logic最新文献_第4页

FORBIDDEN INDUCED SUBGRAPHS AND THE ŁOŚ–TARSKI THEOREM 禁止诱导子图与łoś-tarski定理

The Journal of Symbolic Logic

Pub Date : 2024-01-04 DOI: 10.1017/jsl.2023.99

YIJIA CHEN, JÖRG FLUM

Let $mathscr {C}$ be a class of finite and infinite graphs that is closed under induced subgraphs. The well-known Łoś–Tarski Theorem from classical model theory implies that $mathscr {C}$ is definable in first-order logic by a sentence $varphi $ if and only if $mathscr {C}$ has a finite set of forbidden induced finite subgraphs. This result provides a powerful tool to show nontrivial characterizations of graphs of small vertex cover, of bounded tree-depth, of bounded shrub-depth, etc. in terms of forbidden induced finite subgraphs. Furthermore, by the Completeness Theorem, we can compute from $varphi $ the corresponding forbidden induced subgraphs. This machinery fails on finite graphs as shown by our results:

– There is a class $mathscr {C}$ of finite graphs that is definable in first-order logic and closed under induced subgraphs but has no finite set of forbidden induced subgraphs.
– Even if we only consider classes $mathscr {C}$ of finite graphs that can be characterize

让 $mathscr {C}$ 是一类在诱导子图下封闭的有限图和无限图。经典模型理论中著名的 Łoś-Tarski 定理意味着，当且仅当 $mathscr {C}$ 有一个有限的禁止诱导有限子图集时，$mathscr {C}$ 在一阶逻辑中是可以用一个句子 $varphi $ 来定义的。这一结果提供了一个强大的工具，可以用禁止诱导有限子图来说明小顶点覆盖图、有界树深度图、有界灌木深度图等图的非微观特征。此外，根据完备性定理，我们可以从 $varphi $ 计算出相应的禁止诱导子图。正如我们的结果所示，这一机制在有限图上是失效的：- 有一类$mathscr {C}$ 有限图在一阶逻辑中是可定义的，在诱导子图下是封闭的，但没有有限的禁止诱导子图集。- 即使我们只考虑有限图类$mathscr {C}$，这些有限图可以用有限的禁止诱导子图集来表征，这样的表征也不能从定义$mathscr {C}$的一阶句子$varphi$中计算出来，而且对于任何可计算函数f，表征的大小也不能被$f(|varphi |)$所限定。

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引用次数: 0

LEBESGUE MEASURE ZERO MODULO IDEALS ON THE NATURAL NUMBERS 自然数上的零模理想勒贝格度量

The Journal of Symbolic Logic

Pub Date : 2023-12-29 DOI: 10.1017/jsl.2023.97

VIERA GAVALOVÁ, DIEGO A. MEJÍA

We propose a reformulation of the ideal $mathcal {N}$ of Lebesgue measure zero sets of reals modulo an ideal J on $omega $, which we denote by $mathcal {N}_J$. In the same way, we reformulate the ideal $mathcal {E}$ generated by $F_sigma $ measure zero sets of reals modulo J, which we denote by $mathcal {N}^*_J$. We show that these are $sigma $-ideals and that $mathcal {N}_J=mathcal {N}$ iff J has the Baire property, which in turn is equivalent to $mathcal {N}^*_J=mathcal {E}$

我们提出了一种对理想 $mathcal {N}$ 的重构，即在 $omega $ 上以理想 J 为模量的 Lebesgue 量零实数集，我们用 $mathcal {N}_J$ 表示它。同样，我们重新定义了由$F_sigma $度量为零的实数集 modulo J产生的理想$mathcal {E}$，我们用$mathcal {N}^*_J$ 表示它。我们证明这些是 $sigma $-ideals，并且如果 J 具有 Baire 属性，那么 $mathcal {N}_J=mathcal {N}$，这又等价于 $mathcal {N}^*_J=mathcal {E}$。此外，我们还证明当 J 不具有 Baire 属性时，$mathcal {N}_J$ 不包含共同管理集，并且 $mathcal {N}^*_J$ 包含非管理集。我们还证明了这些ideals modulo J与滤波器（或ideals）的近相干性概念之间的深刻联系。我们还研究了与 $mathcal {N}_J$ 和 $mathcal {N}^*_J$ 相关的心形特征。我们还研究了与 $mathcal {N}_J$ 和 $mathcal {N}^*_J$ 相关的心形特征。我们展示了它们相对于 Cichoń 图的位置，并证明了与连续体的其他非常经典的心形特征相关的一致性结果，只留下了很少的未决问题。为此，我们发现了 $mathrm {add}(mathcal {N})$ 和 $mathrm {cof}(mathcal {N})$ 的新特征。我们还证明，在科恩模型中，我们可以得到与我们的新理想相关的许多不同的红心特征值。

引用次数: 0

THE STRONG AND SUPER TREE PROPERTIES AT SUCCESSORS OF SINGULAR CARDINALS 奇异红衣主教后继者的强树和超树属性

The Journal of Symbolic Logic

Pub Date : 2023-12-22 DOI: 10.1017/jsl.2023.96

WILLIAM ADKISSON

The strong tree property and ITP (also called the super tree property) are generalizations of the tree property that characterize strong compactness and supercompactness up to inaccessibility. That is, an inaccessible cardinal $kappa $ is strongly compact if and only if the strong tree property holds at $kappa $, and supercompact if and only if ITP holds at $kappa $. We present several results motivated by the problem of obtaining the strong tree property and ITP at many successive cardinals simultaneously; these results focus on the successors of singular cardinals. We describe a general class of forcings that will obtain the strong tree property and ITP at the successor of a singular cardinal of any cofinality. Generalizing a result of Neeman about the tree property, we show that it is consistent for ITP to hold at $aleph _n$ for all $2 leq n < omega $ simultaneously with the strong tree property at $aleph _{omega +1}$; we also show that it is consistent for ITP to hold at $aleph _n$ for all

强树性质和ITP（也称作超树性质）是树性质的广义化，它们表征了直到不可访问性为止的强紧凑性和超紧凑性。也就是说，当且仅当强树性质在$kappa $成立时，不可访问的心元$kappa $是强紧凑的；当且仅当ITP在$kappa $成立时，不可访问的心元$kappa $是超紧凑的。我们提出了几个由同时在多个连续红心处获得强树属性和 ITP 问题所激发的结果；这些结果主要集中在奇异红心的后继红心上。我们描述了一类能在任意同频奇异红心的后继处获得强树性质和 ITP 的一般强制。通过推广尼曼关于树属性的一个结果，我们证明了在所有 2 leq n < omega $ 的 $aleph _n$ 处，ITP 与在 $aleph _{omega +1}$ 处的强树属性是一致的；我们还证明了在所有 3 lt; n < omega $ 的 $aleph _n$ 处，ITP 与在 $aleph _{omega +1}$ 处的强树属性是一致的。最后，我们把注意力转向不可数同频的奇异红心，证明强树和超树性质在多个同频奇异的后继处同时成立是一致的。

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引用次数: 0

DEGREE SPECTRA OF HOMEOMORPHISM TYPE OF COMPACT POLISH SPACES 紧凑抛光空间同构类型的度谱

The Journal of Symbolic Logic

Pub Date : 2023-12-11 DOI: 10.1017/jsl.2023.93

MATHIEU HOYRUP, TAKAYUKI KIHARA, VICTOR SELIVANOV

A Polish space is not always homeomorphic to a computably presented Polish space. In this article, we examine degrees of non-computability of presenting homeomorphic copies of compact Polish spaces. We show that there exists a $mathbf {0}'$-computable low$_3$ compact Polish space which is not homeomorphic to a computable one, and that, for any natural number $ngeq 2$, there exists a Polish space $X_n$ such that exactly the high$_{n}$-degrees are required to present the homeomorphism type of $X_n$. Along the way we investigate the computable aspects of Čech homology groups. We also show that no compact Polish space has a least presentation with respect to Turing reducibility.

波兰空间并不总是与可计算呈现的波兰空间同构。在这篇文章中，我们研究了紧凑波兰空间的同构副本的不可计算度。我们证明存在一个$mathbf {0}'$ 可计算的低$_3$紧凑波兰空间，它不与可计算的紧凑波兰空间同构，并且对于任意自然数$ngeq 2$，存在一个波兰空间$X_n$，使得恰好需要高$_{n}$度来呈现$X_n$的同构类型。在此过程中，我们研究了 Čech 同调群的可计算性。我们还证明，就图灵还原性而言，没有一个紧凑波兰空间具有最小呈现。

引用次数: 0

A BOREL MAXIMAL COFINITARY GROUP 最大同余胞群

The Journal of Symbolic Logic

Pub Date : 2023-12-11 DOI: 10.1017/jsl.2023.94

HAIM HOROWITZ, SAHARON SHELAH

We construct a Borel maximal cofinitary group.

我们构建一个 Borel 最大同余元群。

引用次数: 0

DIVIDING LINES BETWEEN POSITIVE THEORIES 积极理论的分界线

The Journal of Symbolic Logic

Pub Date : 2023-12-06 DOI: 10.1017/jsl.2023.89

ANNA DMITRIEVA, FRANCESCO GALLINARO, MARK KAMSMA

We generalise the properties $mathsf {OP}$, $mathsf {IP}$, k-$mathsf {TP}$, $mathsf {TP}_{1}$, k-$mathsf {TP}_{2}$, $mathsf {SOP}_{1}$, $mathsf {SOP}_{2}$, and $mathsf {SOP}_{3}$ to positive logic, and prove various implications and equivalences between them. We also provide a characterisation of stability in positive logic in analogy with the one in full first-order logic, both on the level of formulas and on the level of theories. For simple theories there are the classically equivalent definitions of not having

我们概括了 $mathsf {OP}$, $mathsf {IP}$, k-$mathsf {TP}$, $mathsf {TP}_{1}$, k-$mathsf {TP}_{2}$, $mathsf {SOP}_{1}$ 的性质、$mathsf {SOP}_{2}$ 和 $mathsf {SOP}_{3}$ 到正逻辑，并证明了它们之间的各种蕴涵和等价关系。我们还提供了正逻辑中的稳定性特征，与完全一阶逻辑中的稳定性特征进行了类比，既包括公式层面，也包括理论层面。对于简单理论，我们有经典等价的定义：不具有 $mathsf {TP}$ 和分割具有局部性，我们证明这些定义在正逻辑中也是等价的。最后，我们证明，如果一个厚理论 T 有 $mathsf {IP}$ 或 $mathsf {SOP}_{1}$ ，那么它就有 $mathsf {OP}$ ；如果 T 有 $mathsf {SOP}_{1}$ 或 $mathsf {TP}_{2}$ ，那么它就有 $mathsf {TP}$ 、在前者中，$mathsf {SOP}_{1}$被$mathsf {SOP}$所代替，在后者中，$mathsf {TP}_{1}$被$mathsf {SOP}_{2}$所代替。我们对这最后两个定理的证明是新的，并且利用了金独立性。

引用次数: 0

TWO EXAMPLES CONCERNING EXISTENTIAL UNDECIDABILITY IN FIELDS 关于域中存在不可判性的两个例子

The Journal of Symbolic Logic

Pub Date : 2023-11-23 DOI: 10.1017/jsl.2023.87

PHILIP DITTMANN

We construct an existentially undecidable complete discretely valued field of mixed characteristic with existentially decidable residue field and decidable algebraic part, answering a question by Anscombe–Fehm in a strong way. Along the way, we construct an existentially decidable field of positive characteristic with an existentially undecidable finite extension, modifying a construction due to Kesavan Thanagopal.

我们构造了一个存在上不可判定的完整离散有值混合特征域，它具有存在上可判定的残差域和可判定的代数部分，有力地回答了安斯科姆-费姆提出的一个问题。在此过程中，我们修正了凯萨万-塔纳戈帕尔（Kesavan Thanagopal）的一个构造，构造了一个存在可判的正特征域，它具有存在可判的有限扩展。

引用次数: 0

COFINAL TYPES BELOW Cofinal类型如下

The Journal of Symbolic Logic

Pub Date : 2023-07-24 DOI: 10.1017/jsl.2023.32

ROY SHALEV

It is proved that for every positive integer n, the number of non-Tukey-equivalent directed sets of cardinality

$leq aleph _n$

is at least

$c_{n+2}$

, the

$(n+2)$

-Catalan number. Moreover, the class

$mathcal D_{aleph _n}$

of directed sets of cardinality

$leq aleph _n$

contains an isomorphic copy of the poset of Dyck

$(n+2)$

-paths. Furthermore, we give a complete description whether two successive elements in the copy contain another directed set in between or not.

证明了对于每一个正整数n，基数$leq aleph _n$的非tukey等价有向集的个数至少为$c_{n+2}$，即$(n+2)$ -加泰罗尼亚数。此外，基数$leq aleph _n$的有向集的$mathcal D_{aleph _n}$类包含Dyck $(n+2)$ -paths的偏置集的同构副本。此外，我们给出了副本中两个连续元素之间是否包含另一个有向集的完整描述。

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引用次数: 1

ABELIAN GROUPS DEFINABLE IN p-ADICALLY CLOSED FIELDS 在p-基闭域中可定义的阿贝尔群

The Journal of Symbolic Logic

Pub Date : 2023-07-18 DOI: 10.1017/jsl.2023.52

WILL JOHNSON, NINGYUAN YAO

Recall that a group G has finitely satisfiable generics (fsg) or definable f-generics (dfg) if there is a global type p on G and a small model $M_0$ such that every left translate of p is finitely satisfiable in $M_0$ or definable over $M_0$, respectively. We show that any abelian group definable in a p-adically closed field is an extension of a definably compact fsg definable group by a dfg definable group. We discuss an approach which might prove a similar statement for interpretable abelian groups. In the case where G is an abelian group definable in the standard model $mathbb {Q}_p$, we show that $G^0 = G^{00}$, and that G is an open subgroup of an algebraic group, up to finite factors. This latter result can be seen as a rough classification of abelian definable groups in $mathbb {Q}_p$.

回想一下，如果G上存在一个全局类型p和一个小模型$M_0$，使得p的每一个左平移分别在$M_0$中有限可满足或在$M_0$上可定义，则群G具有有限可满足泛型(fsg)或可定义的f-泛型(dfg)。证明了在p基闭域上任何可定义的阿贝尔群都是可定义紧可定义群由可定义群扩展而来。我们讨论了一种可能证明可解释阿贝尔群的类似陈述的方法。在G是标准模型$mathbb {Q}_p$中可定义的阿贝尔群的情况下，我们证明了$G^0 = G^{00}$，并且G是一个代数群的开子群，最大因子为有限。后一种结果可以看作是$mathbb {Q}_p$中阿贝尔可定义群的粗略分类。

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