On Downey's conjecture
M. M. Arslanov, ISKANDER SH. KALIMULLIN, Steffen Lempp · Journal of Symbolic Logic · 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 degreesf>e>d>0such that any degreeu≤fis either comparable with botheandd, or incomparable with both.