Defining totality in the enumeration degrees

Abstract

We show that if A and B form a nontrivial K-pair, then there is a semi-computable set C such that A ≤e C and B ≤e C. As a consequence, we obtain a definition of the total enumeration degrees: a nonzero enumeration degree is total if and only if it is the join of a nontrivial maximal K-pair. This answers a long-standing question of Hartley Rogers, Jr. We also obtain a definition of the "c.e. in" relation for total degrees in the enumeration degrees. 2010 Mathematics Subject Classification. 03D30. Key words and phrases. enumeration degrees, total enumeration degrees, automorphisms of degree structures.

The jump hierarchy in the enumeration degrees

“…Global definability in the enumeration degrees is connected to its rigidity, in the same manner that we already described for the Turing degrees (see [32]). Cai, Ganchev, Lempp, Miller and Soskova [2] extended the ideas from [14] to give an explanation of this phenomenon. They showed that the total enumeration degrees are first order definable in D e .…”

“…This gives a strong relationship between the automorphism problems of D e and D T : the total enumeration degrees are now a definable automorphism base for the structure of the enumeration degrees, and so the rigidity of D T would implies the the rigidity of D e . Soskova and Slaman [27] used methods from [26] and the results from [2] to show another relationship: the rigidity of any local structure -the structure of the c.e. degrees, D T (≤ 0 T ), and D e (≤ 0 e ), implies the rigidity of D e .…”

The theory of ceers computes true arithmetic

“…Th 2 pNq ď 1 ThpDq, holds as well, thus proving by the Myhill Isomorphism Theorem that the two theories are computably isomorphic, and that ThpDq is as complicated as it can be. The literature here is indeed rich of classical and celebrated results, starting from Simpson [32] who showed that the theory of the Turing degrees is computably isomorphic to Th 2 pNq: see also [33]; to mention two other major reducibilities, the theory of the m-degrees ( [24]), and the theory of the enumeration degrees ( [34]: see also [6]) are also computably isomorphic to Th 2 pNq.…”

Permutations of the Integers Induce Only the Trivial Automorphism of the Turing Degrees

“…Question 1.1 has a long history. Already in 1977, Jockusch and Solovay [11] showed that each jump-preserving automorphism of the Turing degrees is the identity above 0 (4) . Nerode and Shore 1980 [18] showed that each automorphism (not necessarily jump-preserving) is equal to the identity on some cone {a : a ≥ b}.…”

“…The situation for the many-one degrees D m was long known and at the other extreme the same is true for the hyperdegrees D h . Next to be settled may be the enumeration degrees D e , see [4]. For the truth-table degrees D tt some restrictions on possible automorphisms are known.…”