SOP1, SOP2, and antichain tree property

IF 0.6 2区数学 Q2 LOGIC

Annals of Pure and Applied Logic Pub Date : 2023-12-11 DOI:10.1016/j.apal.2023.103402

JinHoo Ahn , Joonhee Kim

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引用次数: 0

Abstract

In this paper, we study some tree properties and their related indiscernibilities. First, we prove that SOP₂ can be witnessed by a formula with a tree of tuples holding ‘arbitrary homogeneous inconsistency’ (e.g., weak k-TP₁ conditions or other possible inconsistency configurations).

And we introduce a notion of tree-indiscernibility, which preserves witnesses of SOP₁, and by using this, we investigate the problem of (in)equality of SOP₁ and SOP₂.

Assuming the existence of a formula having SOP₁ such that no finite conjunction of it has SOP₂, we observe that the formula must witness some tree-property-like phenomenon, which we will call the antichain tree property (ATP, see Definition 4.1). We show that ATP implies SOP₁ and TP₂, but the converse of each implication does not hold. So the class of NATP theories (theories without ATP) contains the class of NSOP₁ theories and the class of NTP₂ theories.

At the end of the paper, we construct a structure whose theory has a formula having ATP, but any conjunction of the formula does not have SOP₂. So this example shows that SOP₁ and SOP₂ are not the same at the level of formulas, i.e., there is a formula having SOP₁, while any finite conjunction of it does not witness SOP₂ (but a variation of the formula still has SOP₂).

查看原文本刊更多论文

SOP1、SOP2 和反链树特性

本文研究了一些树属性及其相关的不可辨性。首先，我们证明了 SOP2 可以由具有 "任意同质不一致性"（如弱 k-TP1 条件或其他可能的不一致性配置）的元组树公式来证明、假定存在一个具有 SOP1 的公式，而它的任何有限连接都不具有 SOP2，我们就会发现这个公式必须具有某种类似树属性的现象，我们称之为反链树属性（ATP，见定义 4.1）。我们证明 ATP 蕴涵 SOP1 和 TP2，但每个蕴涵的反义词都不成立。因此，NATP 理论类（不含 ATP 的理论）包含了 NSOP1 理论类和 NTP2 理论类。在本文的最后，我们构造了一个结构，它的理论有一个具有 ATP 的公式，但该公式的任何连接词都不具有 SOP2。因此，这个例子说明，在公式的层面上，SOP1 和 SOP2 是不一样的，也就是说，有一个公式具有 SOP1，而它的任何有限联结都不具有 SOP2（但该公式的变体仍然具有 SOP2）。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Annals of Pure and Applied Logic 数学-数学

CiteScore

1.40

自引率

12.50%

发文量

审稿时长

200 days

期刊介绍： The journal Annals of Pure and Applied Logic publishes high quality papers in all areas of mathematical logic as well as applications of logic in mathematics, in theoretical computer science and in other related disciplines. All submissions to the journal should be mathematically correct, well written (preferably in English)and contain relevant new results that are of significant interest to a substantial number of logicians. The journal also considers submissions that are somewhat too long to be published by other journals while being too short to form a separate memoir provided that they are of particular outstanding quality and broad interest. In addition, Annals of Pure and Applied Logic occasionally publishes special issues of selected papers from well-chosen conferences in pure and applied logic.