Steffen Lempp
University of Wisconsin-Madison
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Annals of Pure and Applied Logic | 1991
S. Barry Cooper; Leo Harrington; Alistair H. Lachlan; Steffen Lempp; Robert I. Soare
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.
Israel Journal of Mathematics | 1989
S. Barry Cooper; Steffen Lempp; Philip Watson
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.
Proceedings of the American Mathematical Society | 2010
Chitat Chong; Steffen Lempp; Yue Yang
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 .
Advances in Mathematics | 2003
Sergey S. Goncharov; Steffen Lempp
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.
Computability, enumerability, unsolvability | 1996
Marat M. Arslanov; Steffen Lempp; Richard A. Shore
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:
Mathematical Logic Quarterly | 1993
Rodney G. Downey; Steffen Lempp; Richard A. Shore
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.
arXiv: Logic | 2014
Denis R. Hirschfeldt; Carl G. Jockusch; Bjørn Kjos-Hanssen; Steffen Lempp; Theodore A. Slaman
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
Transactions of the American Mathematical Society | 1987
Steffen Lempp
^2_2
Israel Journal of Mathematics | 2003
Rodney G. Downey; Denis R. Hirschfeldt; Steffen Lempp
implies DNR over RCA
Journal of Symbolic Logic | 2014
Uri Andrews; Steffen Lempp; Joseph S. Miller; Keng Meng Ng; Luca San Mauro; Andrea Sorbi
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