Measure and cupping in the Turing degrees

Authors:
George Barmpalias and Andrew E. M. Lewis

Journal:
Proc. Amer. Math. Soc. **140** (2012), 3607-3622

MSC (2010):
Primary 03D28; Secondary 03D10

Published electronically:
February 6, 2012

MathSciNet review:
2929029

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We answer a question of Jockusch by showing that the measure of the Turing degrees that satisfy the cupping property is 0. In fact, every 2-random degree has a strong minimal cover and so fails to satisfy the cupping property.

**[BDN11]**George Barmpalias, Rod Downey, and Keng-Meng Ng.

Jump inversions inside effectively closed sets and applications to randomness.*J. Symbolic Logic*, 2011.

In press.**[Coo04]**S. Barry Cooper,*Computability theory*, Chapman & Hall/CRC, Boca Raton, FL, 2004. MR**2017461****[DH10]**Rodney G. Downey and Denis R. Hirschfeldt,*Algorithmic randomness and complexity*, Theory and Applications of Computability, Springer, New York, 2010. MR**2732288****[DJS96]**Rod Downey, Carl G. Jockusch, and Michael Stob,*Array nonrecursive degrees and genericity*, Computability, enumerability, unsolvability, London Math. Soc. Lecture Note Ser., vol. 224, Cambridge Univ. Press, Cambridge, 1996, pp. 93–104. MR**1395876**, 10.1017/CBO9780511629167.005**[Joc80]**Carl G. Jockusch Jr.,*Degrees of generic sets*, Recursion theory: its generalisation and applications (Proc. Logic Colloq., Univ. Leeds, Leeds, 1979) London Math. Soc. Lecture Note Ser., vol. 45, Cambridge Univ. Press, Cambridge-New York, 1980, pp. 110–139. MR**598304****[Kum00]**Masahiro Kumabe,*A 1-generic degree with a strong minimal cover*, J. Symbolic Logic**65**(2000), no. 3, 1395–1442. MR**1791382**, 10.2307/2586706**[Lew07a]**Andrew E. M. Lewis,*Π₁⁰ classes, strong minimal covers and hyperimmune-free degrees*, Bull. Lond. Math. Soc.**39**(2007), no. 6, 892–910. MR**2392813**, 10.1112/blms/bdm083**[Lew07b]**Andrew E. M. Lewis,*A random degree with strong minimal cover*, Bull. Lond. Math. Soc.**39**(2007), no. 5, 848–856. MR**2365234**, 10.1112/blms/bdm074**[Lew09]**Andrew E. M. Lewis,*Strong minimal covers and a question of Yates: the story so far*, Logic Colloquium 2006, Lect. Notes Log., Assoc. Symbol. Logic, Chicago, IL, 2009, pp. 213–228. MR**2562554**, 10.1017/CBO9780511605321.011**[Mar0s]**D. Martin.

Measure, category, and degrees of unsolvability.

Unpublished manuscript, 1960s.**[ML66]**Per Martin-Löf,*The definition of random sequences*, Information and Control**9**(1966), 602–619. MR**0223179****[Nie09]**André Nies,*Computability and randomness*, Oxford Logic Guides, vol. 51, Oxford University Press, Oxford, 2009. MR**2548883****[Sac63]**Gerald E. Sacks,*Degrees of unsolvability*, Princeton University Press, Princeton, N.J., 1963. MR**0186554****[Soa87]**Robert I. Soare,*Recursively enumerable sets and degrees*, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1987. A study of computable functions and computably generated sets. MR**882921****[Spe56]**Clifford Spector,*On degrees of recursive unsolvability*, Ann. of Math. (2)**64**(1956), 581–592. MR**0082457**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2010):
03D28,
03D10

Retrieve articles in all journals with MSC (2010): 03D28, 03D10

Additional Information

**George Barmpalias**

Affiliation:
Institute for Logic, Language and Computation, Universiteit van Amsterdam 1090 GE, P.O. Box 94242, The Netherlands

Email:
barmpalias@gmail.com

**Andrew E. M. Lewis**

Affiliation:
School of Mathematics, University of Leeds, LS2 9JT Leeds, United Kingdom

Email:
andy@aemlewis.com

DOI:
https://doi.org/10.1090/S0002-9939-2012-11183-9

Received by editor(s):
January 24, 2011

Received by editor(s) in revised form:
March 11, 2011, and April 5, 2011

Published electronically:
February 6, 2012

Additional Notes:
The second author was supported by a Royal Society University Research Fellowship.

Communicated by:
Julia Knight

Article copyright:
© Copyright 2012
American Mathematical Society