CHAPTER 16 - HYPERIMAGINARY FORKING

Summary

Definition 16.1. Let A be a class of hyperimaginaries and let I be a set linearly ordered by <. The sequence of hyperimaginaries (e_i : i ∈ I) is indiscernible over A or it is A-indiscernible if for every n < ω, for every two increasing sequences of indices i₀ < … < i_n and j₀, < … < j_n,. If A is a set, in practice we may always assume that A is a single hyperimaginary. Note that all the hyperimaginaries e_i are in fact of the same sort and hence we can write e_i = [a_i]_E for a single E.

Lemma 16.2. Let d be a hyperimaginary.

1. 1. Let I, J be linearly ordered infinite sets. If (e_i : i ∈ I) is a d-indiscernible sequence of hyperimaginaries, then there is a d-indiscernible sequence (c_i : j ∈ J) such that for every n < ω, for every two increasing sequences of indices i₀ < … < i_n ∈ I and j₀ < … < j_n ∈ J,.

2. 2. If (e_i : i ∈ I) and (d_i : i ∈ I) are d-indiscernible sequences of hyperimaginaries and (e_i : i ∈ I₀) ≡_d (d_i : i ∈ I₀) for each finite subset I₀ ⊆ I, then f((e_i : i ∈ I)) = (d_i : i ∈ I) for some f ∈ Aut(ℭ/d).