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).