Available in electronic format
Available in print format
Bulletin of the American Mathematical Society
Bulletin of the American Mathematical Society
ISSN 1088-9485(e) ISSN 0273-0979(p)
     

The jump is definable in the structure of the degrees of unsolvability

Author(s): S. Barry Cooper
Journal: Bull. Amer. Math. Soc. 23 (1990), 151-158.
MSC (1985): Primary 03D30
MathSciNet review: 1027898
Retrieve article in: PDF

References | Similar articles | Additional information

References:

1.
M. M. Arslanov, Structural properties of the degrees below0’, Dokl. Akad. Nauk. SSSR, (new series) 283 (2) (1985), 270-273.
2.
S. B. Cooper, The strong anticupping property for recursively enumerable degrees, J. Symbolic Logic 54 (1989), 527-539. MR 997886
3.
S. B. Cooper, A jump class of noncappable degrees, J. Symbolic Logic 54 (1989), 324-353. MR 997870
4.
S. B. Cooper and R. L. Epstein, Complementing below recursively enumerable degrees, Ann. Pure Appl. Logic 34 (1987), 15-32. MR 887552
5.
S. B. Cooper, S. Lempp, and P. Watson, Weak density and cupping in the d-r.e. degrees, Israel J. Math. 67 (1989), 137-152. MR 1026559
6.
S. B. Cooper, L. Harrington, A. H. Lachlan, S. Lempp, and R. I. Soare, The d-r.e. degrees are not dense (in preparation).
7.
R. L. Epstein, Degrees of unsolvability: Structure and theory, Lecture Notes in Mathematics No. 759, Springer-Verlag, Berlin, Heidelberg, New York, 1979. MR 551620
8.
L. Feiner, The strong homogeneity conjecture, J. Symbolic Logic 35 (1970), 375-377. MR 286655
9.
L. Harrington and R. A. Shore, Definable degrees and automorphisms of D, Bull. Amer. Math. Soc. (new series) 4 (1981), 97-100. MR 590819
10.
C. G. Jockusch, Jr. and R. A. Shore, Pseudo jump operators I: The R.E. case, Trans. Amer. Math. Soc. 275 (1983), 599-609. MR 682720
11.
C. G. Jockusch, Jr. and R. A. Shore, Pseudo jump operatorsII: Transfinite iterations, hierarchies, and minimal covers, J. Symbolic Logic 49 (1984), 1205-1236. MR 771789
12.
C. G. Jockusch, Jr. and S. G. Simpson, A degree theoretic definition of the ramified analytic hierarchy, Ann. Math. Logic 10 (1976), 1-32. MR 491098
13.
C. G. Jockusch, Jr. and R. M. Solovay, Fixed points of jump preserving automorphisms of degrees, Israel J. Math. 26 (1977), 91-94. MR 432434
14.
S. C. Kleene and E. L. Post, The upper semi-lattice of degrees of recursive unsolvability, Ann. of Math. (2) 59 (1954), 379-407. MR 61078
15.
A. H. Lachlan, A recursively enumerable degree which will not split over all lesser ones, Ann. Math. Logic 9 (1975), 307-365. MR 409150
16.
A. H. Lachlan and R. Lebeuf, Countable initial segments of the degrees of unsolvability, J. Symbolic Logic 41 (1976), 289-300. MR 403937
17.
M. Lerman, Initial segments of the degrees of unsolvability, Ann. of Math. (2) 93 (1971), 365-389. MR 307893
18.
M. Lerman, Degrees of unsolvability, Springer-Verlag, Berlin, Heidelberg, New York City, Tokyo, 1983. MR 708718
19.
A. Nerode and R. A. Shore, Second order logic and first order theories of reducibility orderings, The Kleene Symposium (J. Barwise et al., eds.), North-Holland, Amsterdam, 1980, pp. 181-200. MR 591882
20.
A. Nerode and R. A. Shore, Reducibility orderings: theories, definability and automorphisms, Ann. Math. Logic 18 (1980), 61-89. MR 568916
21.
P. Odifreddi, Classical recursion theory, North-Holland, Amsterdam, New York, Oxford, 1989. MR 982269
22.
D. Posner and R. W. Robinson, Degrees joining to0', J. Symbolic Logic 46(1981), 714-722. MR 641485
23.
E. L.Post, Recursively enumerable sets of positive integers and their decision problems, Bull. Amer. Math. Soc. 50 (1944), 284-316. MR 10514
24.
L. J. Richter, On automorphisms of the degrees that preserve jumps, Israel J.Math. 32(1979), 27-31. MR 531597
25.
H. Rogers, Jr., Theory of recursive functions and effective computability, McGraw-Hill, New York, 1967. MR 224462
26.
R. A. Shore, On homogeneity and definability in the first order theory of the Turing degrees, J. Symbolic Logic 47 (1982), 8-16. MR 644748
27.
R. A. Shore, A non-inversion theorem for the jump operator, Ann. Pure Appl. Logic 40 (1988), 277-303. MR 973483
28.
R. A. Shore, Defining jump classes in the degrees below0', Proc. Amer. Math. Soc. 104 (1988), 287-292. MR 958085
29.
S. G. Simpson, First order theory of the degrees of unsolvability, Ann. of Math. (2) 105 (1977), 121-139. MR 432435
30.
T. A. Slaman and R. A. Shore, Working below a low2 recursively enumerable degree (to appear).
31.
T. A. Slaman and R. A. Shore, Working below a high recursively enumerable degree (in preparation).
32.
T. A. Slaman and J. Steel, Complementation in the Turing degrees, J. Symbolic Logic 54 (1989), 160-176. MR 987329
33.
T. A. Slaman and W. H. Woodin, Definability in the Turing degrees, Illinois J. Math. 30 (1986), 320-334. MR 840131
34.
R. I. Soare, Recursively enumerable sets and degrees, Springer-Verlag, Berlin, Heidelberg, New York, London, 1987. MR 882921
35.
C. E. M. Yates, Initial segments and implications for the structure of degrees, Conference in Mathematical Logic, London, 1970 (W. Hodges, ed.), Springer-Verlag, Berlin, Heidelberg, New York, 1972. MR 357095

Similar Articles:

Retrieve articles in Bulletin of the American Mathematical Society with MSC (1985): 03D30

Retrieve articles in all Journals with MSC (1985): 03D30


Additional Information:

DOI: 10.1090/S0273-0979-1990-15923-3
PII: S 0273-0979(1990)15923-3


  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2008, American Mathematical Society
Privacy Statement
Search the AMSPowered by Google