Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2003.10030

SOP$_1$, SOP$_2$, and antichain tree property

JinHoo Ahn·Joonhee Kim
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper, we study some tree properties and their related indiscernibilities. First, we prove that SOP$_2$ can be witnessed by a formula with a tree of tuples holding 'arbitrary homogeneous inconsistency' (e.g., weak k-TP$_1$ conditions or other possible inconsistency configurations). And we introduce a notion of tree-indiscernibility, which preserves witnesses of SOP$_1$, and by using this, we investigate the problem of (in)equality of SOP$_1$ and SOP$_2$. Assuming the existence of a formula having SOP$_1$ such that no finite conjunction of it has SOP$_2$, 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 SOP$_1$ and TP$_2$, but the converse of each implication does not hold. So the class of NATP theories (theories without ATP) contains the class of NSOP$_1$ theories and the class of NTP$_2$ 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 SOP$_2$. So this example shows that SOP$_1$ and SOP$_2$ are not the same at the level of formulas, i.e., there is a formula having SOP$_1$, while any finite conjunction of it does not witness SOP$_2$ (but a variation of the formula still has SOP$_2$).

Explore related subjects

math.LO

Keep this discovery

BibTeXRIS

JinHoo Ahn, Joonhee Kim. 2020-03-23. SOP$_1$, SOP$_2$, and antichain tree property. https://arxiv.org/abs/2003.10030

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

There is no maximal $K$-degree

The Kolmogorov complexity of a string characterize how complex it is to describe the string. If every prefix of a real $x$ is more complex to describe than every prefix (of the same length) of real $y$, then it is seen as $x$ is more complex to describe than $y$. It is wondered if there is a real $x$ so that no other reals are strictly more complex (to describe) than $x$. The behavior of Kolmogorov complexity functions generated by reals (namely $n\mapsto$ the minimal description length of the real) is quite chaos. Therefore, it is widely believed that there are many reals that are maximally complex to describe. For instance, it is conjectured that all random enough reals have maximal $K$-degree. In this paper, it is shown that there is no real with maximal $K$-degree. Actually, for almost all real $x$, we can uniformly computably find another real whose $K$-degree is strictly above $x$.

math.LO↗
arXiv2026

Quadruples and cubes

We prove, in $\mathsf{ZFC}$, that the $\lambda$-terraced cube relation fails whenever $\lambda$ is an uncountable cardinal. The corresponding terraced relation for quadruples fails for every $\lambda$. If $\lambda$ is $\aleph_0$ then the pretinent terraced relation has consistency strength of at least one Woodin cardinal. We prove positive polarized relations at a successor and a double successor from wondrous ideals. We show, however, that there are no such ideals over two consecutive cardinals simultaneously.

math.LO↗
arXiv2026

Possibilistic Logic over a Logic of Formal Inconsistency

In this article, we have introduced a new possibilistic logic on a logic of formal inconsistency with the aim of developing a possibility theoretic framework to deal with uncertainty and inconsistency meaningfully without leading to a system collapse. We have discussed the syntax and semantics for this logic and have proved the soundness and completeness theorems. A set of new measures of consistency, contradictoriness, and triviality of a set of formulas have been defined. These have then been put to use in an example to show that this framework can provide better means of machine reasoning.

math.LO↗