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December 2007 Maximal chains in the Turing degrees
C. T. Chong, Liang Yu
J. Symbolic Logic 72(4): 1219-1227 (December 2007). DOI: 10.2178/jsl/1203350783

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”.

Citation

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C. T. Chong. Liang Yu. "Maximal chains in the Turing degrees." J. Symbolic Logic 72 (4) 1219 - 1227, December 2007. https://doi.org/10.2178/jsl/1203350783

Information

Published: December 2007
First available in Project Euclid: 18 February 2008

zbMATH: 1131.03017
MathSciNet: MR2371202
Digital Object Identifier: 10.2178/jsl/1203350783

Subjects:
Primary: 03D28 , 03E15 , 03E35 , 03E45

Rights: Copyright © 2007 Association for Symbolic Logic

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Vol.72 • No. 4 • December 2007
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