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

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]**S. Barry Cooper,*The jump is definable in the structure of the degrees of unsolvability*, Bull. Amer. Math. Soc. (N.S.)**23**(1990), no. 1, 151–158. MR**1027898**, 10.1090/S0273-0979-1990-15923-3**[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]**S. Barry Cooper and Richard L. Epstein,*Complementing below recursively enumerable degrees*, Ann. Pure Appl. Logic**34**(1987), no. 1, 15–32. MR**887552**, 10.1016/0168-0072(87)90039-X**[1979]**Richard L. Epstein,*Degrees of unsolvability: structure and theory*, Lecture Notes in Mathematics, vol. 759, Springer, Berlin, 1979. MR**551620****[1999]**Edward R. Griffor (ed.),*Handbook of computability theory*, Studies in Logic and the Foundations of Mathematics, vol. 140, North-Holland Publishing Co., Amsterdam, 1999. MR**1721236****[1984]**Carl G. Jockusch Jr. and Richard A. Shore,*Pseudojump operators. II. Transfinite iterations, hierarchies and minimal covers*, J. Symbolic Logic**49**(1984), no. 4, 1205–1236. MR**771789**, 10.2307/2274273**[1976]**Alistair H. Lachlan,*A recursively enumerable degree which will not split over all lesser ones*, Ann. Math. Logic**9**(1976), no. 4, 307–365. MR**0409150****[1963]**Gerald E. Sacks,*On the degrees less than 0′*, Ann. of Math. (2)**77**(1963), 211–231. MR**0146078****[1985]**Richard A. Shore,*The structure of the degrees of unsolvability*, Recursion theory (Ithaca, N.Y., 1982) Proc. Sympos. Pure Math., vol. 42, Amer. Math. Soc., Providence, RI, 1985, pp. 33–51. MR**791051**, 10.1090/pspum/042/791051**[2000]**Richard A. Shore and Theodore A. Slaman,*Defining the Turing jump*, Math. Res. Lett.**6**(1999), no. 5-6, 711–722. MR**1739227**, 10.4310/MRL.1999.v6.n6.a10**[1939]**Turing, A. M., Systems of logic based on ordinals,*Proc. London Math. Soc. (3)***45**, 161-228.

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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