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Recursion Theory by Liang Yu, Chi Tat Chong

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9.3 Cofinal chains in image

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 AP is a cofinal chain if the members of A are linearly ordered under P and for any zP, there is an xA such that zP 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 〈image, ≤〉 assuming ZFC + CH. The question is considered in this section.

The following fact ...

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