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

Cooper, S. Barry

doi:10.1090/S0273-0979-1990-15923-3

The jump is definable in the structure of the degrees of unsolvability
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Bull. Amer. Math. Soc. 23 (1990), 151-158

References

S. B. Cooper, The strong anticupping property for recursively enumerable degrees, J. Symbolic Logic 54 (1989), no. 2, 527–539. MR 997886, DOI 10.2307/2274867
S. B. Cooper, A jump class of noncappable degrees, J. Symbolic Logic 54 (1989), no. 2, 324–353. MR 997870, DOI 10.2307/2274851
S. Barry Cooper and Richard L. Epstein, Complementing below recursively enumerable degrees, Ann. Pure Appl. Logic 34 (1987), no. 1, 15–32. MR 887552, DOI 10.1016/0168-0072(87)90039-X
S. Barry Cooper, Steffen Lempp, and Philip Watson, Weak density and cupping in the d-r.e. degrees, Israel J. Math. 67 (1989), no. 2, 137–152. MR 1026559, DOI 10.1007/BF02937291
Richard L. Epstein, Degrees of unsolvability: structure and theory, Lecture Notes in Mathematics, vol. 759, Springer, Berlin, 1979. MR 551620, DOI 10.1007/BFb0067135
L. Feiner, The strong homogeneity conjecture, J. Symbolic Logic 35 (1970), 375–377. MR 286655, DOI 10.2307/2270693
Leo Harrington and Richard A. Shore, Definable degrees and automorphisms of ${\cal D}$, Bull. Amer. Math. Soc. (N.S.) 4 (1981), no. 1, 97–100. MR 590819, DOI 10.1090/S0273-0979-1981-14871-0
Carl G. Jockusch Jr. and Richard A. Shore, Pseudojump operators. I. The r.e. case, Trans. Amer. Math. Soc. 275 (1983), no. 2, 599–609. MR 682720, DOI 10.1090/S0002-9947-1983-0682720-1
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, DOI 10.2307/2274273
Carl G. Jockusch Jr. and Stephen G. Simpson, A degree-theoretic definition of the ramified analytical hierarchy, Ann. Math. Logic 10 (1976), no. 1, 1–32. MR 491098, DOI 10.1016/0003-4843(76)90023-1
Carl G. Jockusch Jr. and Robert M. Solovay, Fixed points of jump preserving automorphisms of degrees, Israel J. Math. 26 (1977), no. 1, 91–94. MR 432434, DOI 10.1007/BF03007659
S. C. Kleene and Emil L. Post, The upper semi-lattice of degrees of recursive unsolvability, Ann. of Math. (2) 59 (1954), 379–407. MR 61078, DOI 10.2307/1969708
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 409150, DOI 10.1016/0003-4843(76)90016-4
A. H. Lachlan and R. Lebeuf, Countable initial segments of the degrees of unsolvability, J. Symbolic Logic 41 (1976), no. 2, 289–300. MR 403937, DOI 10.2307/2272227
Manuel Lerman, Initial segments of the degrees of unsolvability, Ann. of Math. (2) 93 (1971), 365–389. MR 307893, DOI 10.2307/1970779
Manuel Lerman, Degrees of unsolvability, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1983. Local and global theory. MR 708718, DOI 10.1007/978-3-662-21755-9
Anil Nerode and Richard A. Shore, Second order logic and first order theories of reducibility orderings, The Kleene Symposium (Proc. Sympos., Univ. Wisconsin, Madison, Wis., 1978), Studies in Logic and the Foundations of Mathematics, vol. 101, North-Holland, Amsterdam-New York, 1980, pp. 181–200. MR 591882
Anil Nerode and Richard A. Shore, Reducibility orderings: theories, definability and automorphisms, Ann. Math. Logic 18 (1980), no. 1, 61–89. MR 568916, DOI 10.1016/0003-4843(80)90004-2
Piergiorgio Odifreddi, Classical recursion theory, Studies in Logic and the Foundations of Mathematics, vol. 125, North-Holland Publishing Co., Amsterdam, 1989. The theory of functions and sets of natural numbers; With a foreword by G. E. Sacks. MR 982269
David B. Posner and Robert W. Robinson, Degrees joining to ${\bf 0}^{\prime }$, J. Symbolic Logic 46 (1981), no. 4, 714–722. MR 641485, DOI 10.2307/2273221
Emil L. Post, Recursively enumerable sets of positive integers and their decision problems, Bull. Amer. Math. Soc. 50 (1944), 284–316. MR 10514, DOI 10.1090/S0002-9904-1944-08111-1
Linda Jean Richter, On automorphisms of the degrees that preserve jumps, Israel J. Math. 32 (1979), no. 1, 27–31. MR 531597, DOI 10.1007/BF02761181
Hartley Rogers Jr., Theory of recursive functions and effective computability, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1967. MR 0224462
Richard A. Shore, On homogeneity and definability in the first-order theory of the Turing degrees, J. Symbolic Logic 47 (1982), no. 1, 8–16. MR 644748, DOI 10.2307/2273376
Richard A. Shore, A noninversion theorem for the jump operator, Ann. Pure Appl. Logic 40 (1988), no. 3, 277–303. MR 973483, DOI 10.1016/0168-0072(88)90034-6
Richard A. Shore, Defining jump classes in the degrees below ${\bf 0}’$, Proc. Amer. Math. Soc. 104 (1988), no. 1, 287–292. MR 958085, DOI 10.1090/S0002-9939-1988-0958085-4
Stephen G. Simpson, First-order theory of the degrees of recursive unsolvability, Ann. of Math. (2) 105 (1977), no. 1, 121–139. MR 432435, DOI 10.2307/1971028
Theodore A. Slaman and John R. Steel, Complementation in the Turing degrees, J. Symbolic Logic 54 (1989), no. 1, 160–176. MR 987329, DOI 10.2307/2275022
Theodore A. Slaman and W. Hugh Woodin, Definability in the Turing degrees, Illinois J. Math. 30 (1986), no. 2, 320–334. MR 840131
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, DOI 10.1007/978-3-662-02460-7
C. E. M. Yates, Initial segments and implications for the structure of degrees, Conference in Mathematical Logic—London ’70 (Proc. Conf., Bedford Coll., London, 1970) Lecture Notes in Math., Vol. 255, Springer, Berlin, 1972, pp. 305–335. MR 0357095