On 1-Degrees Inside m -Degrees

Abstract

Abstract: We study the structure of 1-degrees inside m-degrees and prove that every ∆02m-degree that has more than one 1-degree contains an infinite antichain of 1-degrees. This strengthens Degtev’s result on computably enumerable m-degrees and gives partial answer to the following question stated by Odifreddi: if an m-degree has more than one 1-degree, does it contain an infinite antichain of 1-degrees? The proved result demonstrates that the answer is positive for ∆02m-degrees.