Theories without the tree property of the second kind

Abstract

Abstract. We initiate a systematic study of the class of theories without the tree property of the second kind -NTP 2 . Most importantly, we show: the burden is "sub-multiplicative" in arbitrary theories (in particular, if a theory has TP 2 then there is a formula with a single variable witnessing this); NTP 2 is equivalent to the generalized Kim's lemma and to the boundedness of ist-weight; the dp-rank of a type in an arbitrary theory is witnessed by mutually indiscernible sequences of realizations of the type, after adding some parameters -so the dp-rank of a 1-type in any theory is always witnessed by sequences of singletons; in NTP 2 theories, simple types are co-simple, characterized by the co-independence theorem, and forking between the realizations of a simple type and arbitrary elements satisfies full symmetry; a Henselian valued field of characteristic (0, 0) is NTP 2 (strong, of finite burden) if and only if the residue field is NTP 2 (the residue field and the value group are strong, of finite burden respectively), so in particular any ultraproduct of p-adics is NTP 2 ; adding a generic predicate to a geometric NTP 2 theory preserves NTP 2 .

Generic variations and NTP $$_1$$ 1

“…The idea of the proof is similar as in [3,Theorem 7.3], but we should point out that there seems to be a small gap in the proof from [3]. Namely, in the last paragraph of the proof, since one does not know whether a i j has the same type over acl(b) for various j's, one cannot conclude that there are colorings on acl(a i j b) agreeing on acl(b) (for distinct j's) induced by sending a i0 b 0 to a i j b by an L-elementary map.…”

“…Namely, in the last paragraph of the proof, since one does not know whether a i j has the same type over acl(b) for various j's, one cannot conclude that there are colorings on acl(a i j b) agreeing on acl(b) (for distinct j's) induced by sending a i0 b 0 to a i j b by an L-elementary map. One can, however, work with algebraically closed tuples, which we do below and which also yields a correct proof of [3,Theorem 7.3].…”

Dense codense predicates and the NTP2

“…In this paper, we shall follow the definition given in . (The same notion is called strongly indiscernible array in .) Definition We may view the Cartesian product as a model in the language where < 1 and < 2 are binary relation symbols interpreted in as follows: Given a set of tuples where all have the same arity, …”

Generic trivializations of geometric theories

“…In this section we follow the presentation of NTP 2 theories from . Let T be a complete theory and let be a sufficiently saturated structure.…”