Maximal chains in the Turing degrees

Abstract

We study the problem of existence of maximal chains in the Turing degrees. We show that:

1. $ZF+DC+$“There exists no maximal chain in the Turing degrees” is equiconsistent with $ZFC+$“There exists an inaccessible cardinal”;

2. For all $a \in 2^{\omega}, (\omega_1)^{L[a]} = \omega_1$ if and only if there exists a $\Pi^1_1[a]$ maximal chain in the Turing degrees.

As a corollary, $ZFC$ + “There exists an inaccessible cardinal” is equiconsistent with $ZFC$ + “There is no (bold face) $\underset{\tilde{}}{\Sigma}^1_1$ maximal chain of Turing degrees”.