On Downey's conjecture

Journal of Symbolic Logic 75 (2):401-441 (2010)
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. degrees   n-c.e. degrees   Downey's conjecture
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