Generic sets and minimal 𝛼-degrees

Chong, C. T.

doi:10.1090/S0002-9947-1979-0539912-0

Generic sets and minimal $\alpha$-degrees
HTML articles powered by AMS MathViewer

by C. T. Chong PDF

Trans. Amer. Math. Soc. 254 (1979), 157-169 Request permission

Abstract:

A non-$\alpha$-recursive subset G of an admissible ordinal $\alpha$ is of minimal $\alpha$-degree if every set of strictly lower $\alpha$-degree than that of G is $\alpha$-recursive. We give a characterization of regular sets of minimal $\alpha$-degree below $0’$ 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 $\aleph _\omega ^L$-degree below $0’$.

References

C. T. Chong, An $\alpha$-finite injury method of the unbounded type, J. Symbolic Logic 41 (1976), no. 1, 1–17. MR 476456, DOI 10.2307/2272939
S. B. Cooper, Minimal degrees and the jump operator, J. Symbolic Logic 38 (1973), 249–271. MR 347572, DOI 10.2307/2272061
Wolfgang Maass, On minimal pairs and minimal degrees in higher recursion theory, Arch. Math. Logik Grundlag. 18 (1976), no. 3-4, 169–186. MR 485289, DOI 10.1007/BF02007269
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
G. E. Sacks and S. G. Simpson, The $\alpha$-finite injury method, Ann. Math. Logic 4 (1972), 343–367. MR 369041, DOI 10.1016/0003-4843(72)90004-6
Richard A. Shore, Minimal $\alpha$-degrees, Ann. Math. Logic 4 (1972), 393–414. MR 369042, DOI 10.1016/0003-4843(72)90006-X
Richard A. Shore, Splitting an $\alpha$-recursively enumerable set, Trans. Amer. Math. Soc. 204 (1975), 65–77. MR 379154, DOI 10.1090/S0002-9947-1975-0379154-1
Clifford Spector, On degrees of recursive unsolvability, Ann. of Math. (2) 64 (1956), 581–592. MR 82457, DOI 10.2307/1969604
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) North-Holland, Amsterdam, 1970, pp. 63–83. MR 0269505