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Bounding minimal degrees by computably enumerable degrees
Published online by Cambridge University Press: 12 March 2014
Abstract
In this paper, we prove that there exist computably enumerable degrees a and b such that a > b and for any degree x, if x ≤ a and x is a minimal degree, then x < b.
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- Copyright © Association for Symbolic Logic 1998
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REFERENCES
[1]
Arslanov, M. M., Structural properties of the degrees below 0′, Doklady Akademii Nauk UzSSR, vol. 283 (1985).Google Scholar
[2]
Arslanov, M. M., Lempp, S., and Shore, R. A., On isolating and isolated d-r.e. degrees, (Cooper, S. B., Siaman, T. A., and Wainer, S. S., editors), London Mathematical Society Lecture Note Series, vol. 224, Cambridge University Press, 1996.Google Scholar
[3]
Cooper, S. B., Beyond Gödel's theorem: The failure to capture information content, to appear.Google Scholar
[4]
Cooper, S. B., Local degree theory, Handbook of computability theory (Griffor, E., editor), North-Holland, to appear.Google Scholar
[5]
Cooper, S. B., Minimal degrees and the jump operators, this Journal, vol. 38 (1973), pp. 249—271.Google Scholar
[6]
Cooper, S. B., Some negative results on minimal degrees below O′, Recursive Function Theory Newsletter, vol. 34 (1986), item 353 (abstract).Google Scholar
[7]
Cooper, S. B., The strong anti-cupping property for recursively enumerable degrees, this Journal, vol. 54 (1989), pp. 527—539.Google Scholar
[8]
Cooper, S. B., The jump is definable in the structure of the degrees of unsolvability, Bulletin of the American Mathematical Society, vol. 23 (1990), pp. 151—158.Google Scholar
[9]
Cooper, S. B., Definability and global degree theory, Logic colloquium '90 (Oikkonen, J. and Väänänen, J., editors), Lecture Notes in Logic, vol. 2, Springer-Verlag, Berlin, Heidelberg, New York, 1993, pp. 25—45.Google Scholar
[10]
Cooper, S. B., Discontinuous phenomenon and Turing definability, Proceedings of the international conference of algebra and analysis, 1994, Kazan, 06 1994.Google Scholar
[11]
Cooper, S. B., Rigidity and definability in the noncomputable universe, Proceedings of the 9th international congress of logic, methodology and philosophy of science (Prawitz, D., Skyrms, B., and Westerstahl, D., editors), North-Holland, Amsterdam, 1994, pp. 209—236.Google Scholar
[12]
Cooper, S. B., On a conjecture of Kleene and Post, Complexity, logic and recursion theory (Sorbi, A., editor), Lecture Notes in Pure and Applied Mathematics, vol. 187, Marcel Dekker, New York, 1997, pp. 93—122.Google Scholar
[13]
Cooper, S. B. and Epstein, R. L., Complementing below recursively enumerable degrees, Annals of Pure and Applied Logic, vol. 34 (1987), pp. 15—32.Google Scholar
[16]
Ding, D. and Qian, L., Isolated d.r.e. degrees are dense in r.e. degrees, to appear.Google Scholar
[18]
Downey, R. G., Δ2
0 degrees and transfer theorems, Illinois Journal of Mathematics, vol. 31 (1987), pp. 419—427.Google Scholar
[19]
Downey, R. G., D-r.e. degrees and the nondiamond theorem, Bulletin of the London Mathematical Society, vol. 21 (1989), pp. 43—50.Google Scholar
[20]
Epstein, R. L., Minimal degrees of unsolvability and the full approximation construction, vol. 3, Memoirs of the American Mathematical Society, no. 162, American Mathematical Society, Providence, R.I., 1975.Google Scholar
[21]
Epstein, R. L., Degrees of unsolvability: Structure and theory, Lecture Notes in Mathematics, vol. 759, Springer-Verlag, Berlin, Heidelberg, New York, 1979.Google Scholar
[22]
Epstein, R. L., Initial segments of degrees below 0′, vol. 30, Memoirs of the American Mathematical Society, no. 241, American Mathematical Society, Providence, R.I., 1981.Google Scholar
[23]
Friedberg, R. M., Two recursively enumerable sets of incomparable degrees of unsolvability, Proceedings of the National Academy of Sciences, USA, vol. 43 (1957), pp. 236—238.CrossRefGoogle ScholarPubMed
[24]
Jiang, Z., Diamond lattice embedded into d.r.e. degrees, Science in China, vol. 36 (1993), pp. 803—811.Google Scholar
[25]
Jockusch, C. G. Jr., Simple proofs of some theorems on high degrees, Canadian Journal of Mathematics, vol. 29 (1977), pp. 1072—1080.Google Scholar
[26]
Jockusch, C. G. Jr., Degrees of generic sets, Recursion theory: Its generalizations and applications (Drake, F. R. and Wainer, S. S., editors), London Mathematical Society Lecture Notes Series, vol. 45, Cambridge University Press, Cambridge, New York, Melbourne, 1980, Proceedings of Logic Colloquium '79, Leeds, 08 1979, pp. 110—139.Google Scholar
[27]
Kleene, S. C. and Post, E. L., The upper semi-lattice of degrees of recursive unsolvability, Annals of Mathematics, vol. 2 (1954), no. 59, pp. 379–407.Google Scholar
[28]
Lachlan, A. H., Bounding minimal pairs, this Journal, vol. 44 (1979), pp. 626—642.Google Scholar
[29]
LaForte, G., Phenomena in the n-r.e. and n-REA degrees,
Ph.D. thesis
. University of Michigan, 1995.Google Scholar
[30]
Lerman, M., Degrees of unsolvability, Perspectives in Mathematical Logic, Omega Series, Springer-Verlag, Berlin, Heidelberg, London, New York, Tokyo, 1983.Google Scholar
[31]
Lerman, M., Degrees which do not bound minimal degrees, Annals of Pure and Applied Logic, vol. 30 (1986), pp. 249—276.Google Scholar
[33]
Li, A., External center theorem of the recursively enumerable degrees, to appear.Google Scholar
[34]
Li, A. and Yi, X., Cupping the recursively enumerable degrees by d.r.e. degrees, to appear in Proceedings of the London Mathematical Society.Google Scholar
[35]
Posner, D. B., The upper semilattice of degrees ≤ 0′ is complemented, this Journal, vol. 46 (1981), pp. 705—713.Google Scholar
[36]
Posner, D. B. and Robinson, R. W., Degrees joining to 0′, this Journal, vol. 46 (1981), pp. 714—722.Google Scholar
[37]
Post, E. L., Degrees of recursive unsolvability: Preliminary report, Bulletin of the American Mathematical Society, vol. 54 (1948), pp. 641—642, abstract.Google Scholar
[38]
Sacks, G. E., A minimal degree less than 0′, Bulletin of the American Mathematical Society, vol. 67 (1961), pp. 416—419.Google Scholar
[39]
Sacks, G. E., Forcing with perfect closed sets, Axiomatic set theory I (Scott, D., editor), Proceedings of Symposia in Pure Mathematics, Los Angeles, 1967, American Mathematical Society, Providence, R.I., 1971, pp. 331—355.Google Scholar
[40]
Sasso, L. P., A cornucopia of minimal degrees, this Journal, vol. 395 (1970), pp. 383—388.Google Scholar
[41]
Seetapun, D. and Slaman, T. A., Minimal complements, unpublished manuscript, 1992.Google Scholar
[42]
Shoenfield, J. R., A theorem on minimal degrees, this Journal, vol. 31 (1966), pp. 539—544.Google Scholar
[43]
Shore, R. A., Defining jump classes in the degrees below 0′, Proceedings of the American Mathematical Society, vol. 104 (1988), pp. 287—292.Google Scholar
[44]
Slaman, T. A., A recursively enumerable degree that is not the top of a diamond in the Turing degrees, to appear.Google Scholar
[45]
Slaman, T. A., The recursively enumerable degrees as a substructure of the Δ2
0 degrees, handwritten notes, 10
1983.Google Scholar
[46]
Slaman, T. A. and Steel, J. R., Complementation in the Turing degrees, this Journal, vol. 54 (1989), pp. 160—176.Google Scholar
[47]
Slaman, T. A. and Woodin, W. H., Definability in the Turing degrees, Illinois Journal of Mathematics, vol. 30 (1986), pp. 320—334.Google Scholar
[48]
Soare, R. I., Recursively enumerable sets and degrees, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, Heidelberg, London, New York, Paris, Tokyo, 1987.Google Scholar
[50]
Spector, C., On degrees of recursive unsolvability, Annals of Mathematics, vol. 2 (1956), no. 64, pp. 581—592.Google Scholar
[51]
Turing, A. M., Systems of logic based on ordinals, Proceedings of the London Mathematical Society, vol. 45 (1939), pp. 161—228, reprinted in Davis [1965], pp. 154–222.Google Scholar
[52]
Yates, C. E. M., Initial segments of the degrees of unsolvability, part II: minimal degrees, this Journal, vol. 35 (1970), pp. 243—266.Google Scholar