Generic sets and minimal degrees
Author:
C. T. Chong
Journal:
Trans. Amer. Math. Soc. 254 (1979), 157169
MSC:
Primary 03D60; Secondary 03D30
MathSciNet review:
539912
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: A nonrecursive subset G of an admissible ordinal is of minimal degree if every set of strictly lower degree than that of G is recursive. We give a characterization of regular sets of minimal degree below via the notion of genericity. We then apply this to outline some 'minimum requirements' to be satisfied by any construction of a set of minimal degree below .
 [1]
C.
T. Chong, An 𝛼finite injury method of the unbounded
type, J. Symbolic Logic 41 (1976), no. 1,
1–17. MR
0476456 (57 #16019)
 [2]
S.
B. Cooper, Minimal degrees and the jump operator, J. Symbolic
Logic 38 (1973), 249–271. MR 0347572
(50 #75)
 [3]
Wolfgang
Maass, On minimal pairs and minimal degrees in higher recursion
theory, Arch. Math. Logik Grundlagenforsch. 18
(1976), no. 34, 169–186. MR 0485289
(58 #5136)
 [4]
Gerald
E. Sacks, Forcing with perfect closed sets, Axiomatic Set
Theory (Proc. Sympos. Pure Math., Vol. XIII, Part I, Univ. California, Los
Angeles, Calif., 1967) Amer. Math. Soc., Providence, R.I., 1971,
pp. 331–355. MR 0276079
(43 #1827)
 [5]
G.
E. Sacks and S.
G. Simpson, The 𝛼finite injury method, Ann. Math.
Logic 4 (1972), 343–367. MR 0369041
(51 #5277)
 [6]
Richard
A. Shore, Minimal 𝛼degrees, Ann. Math. Logic
4 (1972), 393–414. MR 0369042
(51 #5278)
 [7]
Richard
A. Shore, Splitting an 𝛼recursively
enumerable set, Trans. Amer. Math. Soc. 204 (1975), 65–77.
MR
0379154 (52 #60), http://dx.doi.org/10.1090/S00029947197503791541
 [8]
Clifford
Spector, On degrees of recursive unsolvability, Ann. of Math.
(2) 64 (1956), 581–592. MR 0082457
(18,552d)
 [9]
C.
E. M. Yates, Initial segments of the degrees of unsolvability. I. A
survey, Mathematical Logic and Foundations of Set Theory (Proc.
Internat. Colloq., Jerusalem, 1968) NorthHolland, Amsterdam, 1970,
pp. 63–83. MR 0269505
(42 #4400)
 [1]
 C. T. Chong, An finite injury method of the unbounded type, J. Symbolic Logic 41 (1976), 117. MR 0476456 (57:16019)
 [2]
 S. B. Cooper, Minimal degrees and the jump operator, J. Symbolic Logic 38 (1973), 249271. MR 0347572 (50:75)
 [3]
 W. Maass, On minim pairs and minimal degrees in higher recursion theory, Arch. Math. Logik Grundlagenforsch. 18 (1976), 169186. MR 0485289 (58:5136)
 [4]
 G. E. Sacks, Forcing with perfect closed sets, Proc. Sympos. Pure Math., vol. 13, Amer. Math. Soc., Providence, R. I., 1971. pp. 331355. MR 0276079 (43:1827)
 [5]
 G. E. Sacks and S. G Simpson, The finite injury method, Ann. Math. Logic 4 (1972), 343367. MR 0369041 (51:5277)
 [6]
 R. A. Shore, Minimal degrees, Ann. Math. Logic 4 (1972), 393414. MR 0369042 (51:5278)
 [7]
 , Splitting an recursively enumerable set, Trans. Amer. Math. Soc. 204 (1975), 6578. MR 0379154 (52:60)
 [8]
 C. Spector, On degrees of recursive unsolvability, Ann. of Math. (2) 64 (1956), 581592. MR 0082457 (18:552d)
 [9]
 C. E. M. Yates, Initial segments of the degrees of unsolvability, Mathematical Logic and Foundations of Set Theory (Y. BarHillel, Editor), NorthHolland, Amsterdam, 1970, pp. 6383. MR 0269505 (42:4400)
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC:
03D60,
03D30
Retrieve articles in all journals
with MSC:
03D60,
03D30
Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197905399120
PII:
S 00029947(1979)05399120
Keywords:
Admissible ordinal,
minimal degree,
generic set
Article copyright:
© Copyright 1979
American Mathematical Society
