Martin's conjecture in the enumeration degrees

Abstract: Martin's conjecture is a long open problem that seeks to prove the empirical observation that "naturally occurring" Turing degrees are well-ordered. The conjecture posits that the only natural constructions of incomputable degrees arise from iterations of the Turing jump. Even though the full conjecture remains open, several significant partial results have been obtained both in the Turing degrees and by translating the conjecture to other degree structures.

The study of the enumeration degrees has gained relevance in recent years for their applications to effective mathematics and for their structural connections to the Turing degrees. In this setting, Martin's conjecture is relevant due to the existence of a definable copy of the Turing degrees inside the enumeration degrees and two natural operations that extend the Turing jump: the enumeration jump and the skip. However, the unique features of the enumeration degrees pose challenges to even formulating an analogue to Martin's conjecture.

I will present a surprising positive result based on Bard's local approach to the uniform Martin's conjecture. From this, we can prove part 1 of Martin's conjecture for uniformly Turing-to-enumeration invariant functions. Additionally, I discuss several counterexamples, including an invariant function in the enumeration degrees that fails to be uniformly invariant.

logic

Audience: researchers in the topic

Online logic seminar

Series comments: Description: Seminar on all areas of logic