### 9.3 Cofinal chains in

#### 9.3.1 Existence of a cofinal maximal chain of order type ω1

Definition 9.3.1. Given a partial ordering 〈P, <P〉, a set A ⊆ P is a cofinal chain if the members of A are linearly ordered under P and for any z ∈ P, there is an x ∈ A such that z ≤P x.

It is not difficult to see that under ZFC, there is a cofinal chain of Turing degrees if and only if CH holds. A natural question is whether there exists a cofinal maximal chain of order type ω1 in 〈, ≤〉 assuming ZFC + CH. The question is considered in this section.

The following fact ...