Steffen Lempp

The d.r.e. degrees are not dense

Abstract By constructing a maximal incomplete d.r.e. degree, the nondensity of the partial order of the d.r.e. degrees is established. An easy modification yields the nondensity of the n -r.e. degrees ( n ⩾2) and of the ω-r.e. degrees.

Weak density and cupping in the d-r.e. degrees

Consider the Turing degrees of differences of recursively enumerable sets (the d-r.e. degrees). We show that there is a properly d-r.e. degree (a d-r.e. degree that is not r.e.) between any two comparable r.e. degrees, and that given a high r.e. degreeh, every nonrecursive d-r.e. degree ≦h cups toh by a low d-r.e. degree.

On the role of the collection principle for Sigma^0_2-formulas in second-order reverse mathematics

We show that the principle PART from Hirschfeldt and Shore is equivalent to the ∑0 2 -Bounding principle B∑ 0 2 over RCA 0 , answering one of their open questions. Furthermore, we also fill a gap in a proof of Cholak, Jockusch and Slaman by showing that D 2 2 implies B∑ 0 2 and is thus indeed equivalent to Stable Ramseys Theorem for Pairs (SRT 2 2 ). This also allows us to conclude that the combinatorial principles IPT 2 2 , SPT 2 2 and SIPT 2 2 defined by Dzhafarov and Hirst all imply B∑ 0 2 and thus that SPT 2 2 and SIPT 2 2 are both equivalent to SRT 2 2 as well. Our proof uses the notion of a bi-tame cut, the existence of which we show to be equivalent, over RCA 0 , to the failure of B∑ 0 2 .

The computable dimension of ordered abelian groups

Abstract Let G be a computable ordered abelian group. We show that the computable dimension of G is either 1 or ω , that G is computably categorical if and only if it has finite rank, and that if G has only finitely many Archimedean classes, then G has a computable presentation which admits a computable basis.

On isolating r.e. and isolated d-r.e. degrees

The notion of a recursively enumerable (r.e.) set, i.e. a set of integers whose members can be e ectively listed, is a fundamental one. Another way of approaching this de nition is via an approximating function fAsgs2! to the set A in the following sense: We begin by guessing x = 2 A at stage 0 (i.e. A0(x) = 0); when x later enters A at a stage s+1, we change our approximation from As(x) = 0 to As+1(x) = 1. Note that this approximation (for xed) x may change at most once as s increases, namely when x enters A. An obvious variation on this de nition is to allow more than one change: A set A is 2r.e. (or d-r.e.) if for each x, As(x) change at most twice as s increases. This is equivalent to requiring the set A to be the di erence of two r.e. sets A1 A2. (Similarly, one can de ne n-r.e. sets by allowing at most n changes for each x.) The notion of d-r.e. and n-r.e. sets goes back to Putnam [1965] and Gold [1965] and was investigated (and generalized) by Ershov [1968a, b, 1970]. Cooper showed that even in the Turing degrees, the notions of r.e. and dr.e. di er:

Highness and bounding minimal pairs

We show the existence of a high r. e. degree bounding only joins of minimal pairs and of a high2 nonbounding r. e. degree. MSC: 03D25.

The Strength of Some Combinatorial Principles Related to Ramsey's Theorem for Pairs

We study the reverse mathematics and computability-the\-o\-re\-tic strength of (stable) Ramseys Theorem for pairs and the related principles COH and DNR. We show that SRT

Hyperarithmetical index sets in recursion theory

^2_2

Computability-theoretic and proof-theoretic aspects of partial and linear orderings

implies DNR over RCA

UNIVERSAL COMPUTABLY ENUMERABLE EQUIVALENCE RELATIONS

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