Positive indiscernibles

Abstract

We generalise various theorems for finding indiscernible trees and arrays to positive logic: based on an existing modelling theorem for s-trees, we prove modelling theorems for str-trees, str\(_0\)-trees (the reduct of str-trees that forgets the length comparison relation) and arrays. In doing so, we prove stronger versions for basing—rather than locally basing or EM-basing—str-trees on s-trees and str\(_0\)-trees on str-trees. As an application we show that a thick positive theory has k-\(\mathsf {TP_2}\) iff it has 2-\(\mathsf {TP_2}\)