On Downey's conjecture

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.

• Type: Journal
• Year: 2010
• Where: JSYML
• Authors: Marat M. Arslanov, Iskander Sh. Kalimullin, Steffen Lempp
• Keywords: Degree Structures, Elementarily Equivalent, JSYML 2010, Refuting