A splitting theorem for degrees

Authors:
Richard A. Shore and Theodore A. Slaman

Journal:
Proc. Amer. Math. Soc. **129** (2001), 3721-3728

MSC (2000):
Primary 03D25, 03D30; Secondary 03D55

DOI:
https://doi.org/10.1090/S0002-9939-01-06015-4

Published electronically:
April 25, 2001

MathSciNet review:
1860508

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We prove that, for any , and with and r.e., in , there are pairs and such that ; ; and, for any and from and any set , if and , then . We then deduce that for any degrees , , and such that and are recursive in , , and is into , can be split over avoiding . This shows that the Main Theorem of Cooper (Bull. Amer. Math. Soc. **23** (1990), 151-158) is false.

**[1990]**Cooper, S. B., The jump is definable in the structure of the degrees of unsolvability (research announcement),*Bull. Amer. Math. Soc.***23**, 151-158. MR**91e:03044****[1993]**Cooper, S. B., On a conjecture of Kleene and Post, Department of Pure Mathematics, Leeds University, 1993 Preprint Series No. 7.**[2000]**Cooper, S. B., The Turing definability of the relation of ``computably enumerable in'', Computability Theory Seminar, University of Leeds.**[1987]**Cooper, S. B. and Epstein, R. L., Complementing below recursively enumerable degrees,*Ann. Pure Appl. Logic***34**, 15-32. MR**89f:03035****[1979]**Epstein, R.,*Degrees of Unsolvability: Structure and Theory*,*Lecture Notes in Mathematics***759**, Springer-Verlag, Heidelberg. MR**81e:03042****[1999]**Griffor, E. (ed.),*Handbook of Computability Theory*, North-Holland, Amsterdam. MR**2000e:03001****[1984]**Jockusch, C. G. Jr. and Shore, R. A., Pseudo-jump operators II: transfinite iterations, hierarchies and minimal covers,*J. Symb. Logic***49**, 1205-1236. MR**86g:03072****[1976]**Lachlan, A., A recursively enumerable degree which will not split over all lesser ones,*Ann. Math. Logic***9**, 307-365. MR**53:12912****[1963]**Sacks, G. E., On the degrees less than ,*Ann. of Math. (2)***77**, 211-231. MR**26:3604****[1985]**Shore, R. A., The structure of the degrees of unsolvability, in*Recursion theory*, Proc. Symp. Pure Math.**42**, A. Nerode and R. A. Shore eds., Amer. Math. Soc., Providence RI, 33-51. MR**87e:03100****[2000]**Shore, R. A. and Slaman, T. A., Defining the Turing jump,*Math. Res. Lett.***6**(1999), 711-722. MR**2000m:03104****[1939]**Turing, A. M., Systems of logic based on ordinals,*Proc. London Math. Soc. (3)***45**, 161-228.

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
03D25,
03D30,
03D55

Retrieve articles in all journals with MSC (2000): 03D25, 03D30, 03D55

Additional Information

**Richard A. Shore**

Affiliation:
Department of Mathematics, Cornell University, Ithaca, New York 14853

Email:
shore@math.cornell.edu

**Theodore A. Slaman**

Affiliation:
Department of Mathematics, University of California, Berkeley, California 94720-3840

Email:
slaman@math.berkeley.edu

DOI:
https://doi.org/10.1090/S0002-9939-01-06015-4

Keywords:
Degrees,
Turing degrees,
recursively enumerable degrees,
splitting theorems

Received by editor(s):
November 15, 1999

Received by editor(s) in revised form:
May 4, 2000

Published electronically:
April 25, 2001

Additional Notes:
The first author was partially supported by NSF Grant DMS-9802843.

The second author was partially supported by NSF Grant DMS-97-96121.

Communicated by:
Carl G. Jockusch, Jr.

Article copyright:
© Copyright 2001
American Mathematical Society