SOP1, SOP2, and antichain tree property

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).