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On Downey's conjecture

Published online by Cambridge University Press:  12 March 2014

Marat M. Arslanov ,
Iskander Sh. Kalimullin  and
Steffen Lempp
Marat M. Arslanov
Affiliation:
Department of Mathematics, Kazan State University, Ul. Kremlevskaya 18, 420008 Kazan, Russia. E-mail: Marat.Arslanov@ksu.ru
Iskander Sh. Kalimullin
Affiliation:
Department of Mathematics, Kazan State University, Ul. Kremlevskaya 18, 420008 Kazan, Russia. E-mail: Iskander.Kalimullin@ksu.ru
Steffen Lempp
Affiliation:
Department of Mathematics, University of Wisconsin, Madison. WI 53706-1388, USA. E-mail: lempp@math.wisc.edu, URL: http://www.math.wisc.edu/~lempp
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Abstract

We prove that the degree structures of the d.c.e. and the 3-c.e. Turing degrees are not elementarily equivalent, thus refuting a conjecture of Downey. More specifically, we show that the following statement fails in the former but holds in the latter structure: There are degrees f > e > d > 0 such that any degree u ≤ f is either comparable with both e and d, or incomparable with both.

Keywords

d.c.e. degreesn-c.e. degreesDowney's conjecture
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 75 , Issue 2 , June 2010 , pp. 401 - 441
Copyright
Copyright © Association for Symbolic Logic 2010

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