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Definability and global degree theory

Published online by Cambridge University Press:  24 March 2017

S. Barry Cooper
Affiliation:
University of Leeds
Juha Oikkonen
Affiliation:
University of Helsinki
Jouko Väänänen
Affiliation:
University of Helsinki
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Publisher: Cambridge University Press
Print publication year: 2017

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References

Cooper, S. B., On a conjecture of Kleene and Post, to appear.
Cooper, S. B., The recursively enumerable degrees are absolutely definable, to appear.
Enderton, H. B. and Putnam, H., A note on the hyperarithmetical hierarchy, J. Symbolic Logic 35 (1970), 429–430.Google Scholar
Epstein, R. L., “Degrees of Unsolvability. Structure and Theory,” Lecture Notes in Mathematics No. 759, Springer-Verlag, New York, 1979.
Feiner, L., The strong homogeneity conjecture, J. Symbolic Logic 35 (1970), 375–377.Google Scholar
Friedberg, R. M., A criterion for completeness of degrees of unsolvability , J. Symbolic Logic 22 (1957), 159–160.Google Scholar
Gödel, K., On undecidable propositions of formal mathematical systems, mimeographed notes, in “The Undecidable. Basic Papers on Undecidable Propositions, Unsolvable Problems, and Computable Functions,” (Davis, M., ed.), Raven Press, New York, 1965, pp. 39–71.
Harrington, L. and Kechris, A., A basis result for Σ3 0 sets of reals with an application to minimal covers, Proc. Amer. Math. Soc. 53 (1975), 445–448.Google Scholar
Harrington, L. and Shore, R. A., Definable degrees and automorphisms of D , Bull. Amer. Math. Soc. (new series) 4 (1981), 97–100.Google Scholar
Jockusch, C. G. Jr., Degrees of generic sets, Lond. Math. Soc. Lect. Notes 45 (1981), 110–139.Google Scholar
Jockusch, C. G. Jr. and Posner, D., Automorphism bases for degrees of unsolvability, Israel J. Math. 40 (1981), 150–164.Google Scholar
Jockusch, C. G. Jr. and Shore, R. A., Pseudo jump operators II: Transfinite iterations, hierarchies, and minimal covers, J. Symbolic Logic 49 (1984), 1205—1236.Google Scholar
Jockusch, C. G. Jr. and Simpson, S. G., A degree theoretic definition of the ramified analytic hierarchy, Ann. Math. Logic 10 (1976), 1–32.Google Scholar
Jockusch, C. G. Jr. and Soare, R. I., Minimal covers and arithmetical sets, Proc. Amer. Math. Soc. 25 (1970), 856–859.Google Scholar
Jockusch, C. G. Jr. and Solovay, R. M., Fixed points of jump preserving automorphisms of degrees, Israel J. Math. 26 (1977), 91–94.Google Scholar
Kleene, S. C., General recursive functions of natural numbers, Math. Ann. 112 (1936), 727–742.Google Scholar
Kleene, S. C., Recursive predicates and quantifiers, Trans. Amer. Math. Soc. 53 (1943), 41–73.Google Scholar
Kleene, S. C. and Post, E. L., The upper semi-lattice of degrees of recursive unsolvability, Ann. Math. (2) 59 (1954), 379–407.Google Scholar
Lachlan, A. H., Distributive initial segments of the degrees of unsolvability, Z. Math. Logik Grundlag. Math. 14 (1968), 457–472.Google Scholar
Lachlan, A. H., A recursively enumerable degree which will not split over all lesser ones, Ann. Math. Logic 9 (1975), 307–365.Google Scholar
MacIntyre, J. M., Transfinite extensions of Friedberg's completeness criterion, J. Symbolic Logic 42 (1977), 1–10.Google Scholar
Nerode, A. and Shore, R. A., Second order logic and first order theories of reducibility orderings, in “The Kleene Symposium,” (Barwise, J. et al., eds.), North-Holland, Amsterdam, 1980, pp. 181–200.
Nerode, A. and Shore, R. A., Reducibiliiy orderings: theories, definability and automorphisms, Ann. Math. Logic 18 (1980), 61–89.Google Scholar
Odifreddi, P., “Classical Recursion Theory,” North-Holland, Amsterdam, New York, Oxford, 1989.
Odifreddi, P. and Shore, R. A., Global properties of local structures of degrees, to appear.
Posner, D. and Robinson, R. W., Degrees joining to 0′, J. Symbolic Logic 46 (1981), 714–722.Google Scholar
Post, E. L., Recursively enumerable sets of positive integers and their decision problems, Bull. Amer. Math. Soc. 50 (1944), 284–316.Google Scholar
Post, E. L., Degrees of recursive unsolvability , Bull. Amer. Math. Soc. 54 (1948), 641–642.Google Scholar
Richter, L. J., On automorphisms of the degrees that preserve jumps, Israel J. Math. 32 (1979), 27–31.Google Scholar
Rogers, H. Jr., “Theory of recursive functions and effective computability,” McGraw-Hill, New York, 1967.
Sacks, G. E., Forcing with perfect closed sets, in “Axiomatic Set Theory I,” (Scott, D., ed.), Proc. Symp. Pure Math. 13, Los Angeles, 1967, Amer. Math. Soc., Providence, R.I., 1971, pp. 331–355.
Sacks, G. E., Some open questions in recursion theory, in “Recursion Theory Week,” (Ebbinghaus, H. D. et al., eds.), Lecture Notes in Mathematics No. 1141, Springer-Verlag, Berlin, Heidelberg, New York, 1985, pp. 333–342.
Shore, R. A., The homogeneity conjecture, Proc. Natl. Acad. Sci. U.S.A. 76 (1979), 4218–4219.Google Scholar
Shore, R. A., The degrees of unsolvability: global results, in “Logic Year 1979–80: University of Connecticut,” (eds. Lerman, M. et al.), Lecture Notes in Mathematics No. 859, Springer-Verlag, Berlin, Heidelberg, New York, 1981, pp. 283–301.
Shore, R. A., On homogeneity and definability in the first order theory of the Turing degrees, J. Symbolic Logic 47 (1982), 8–16.Google Scholar
Shore, R. A., Defining jump classes in the degrees below 0′, Proc. Amer. Math. Soc. 104 (1988), 287–292.Google Scholar
Shore, R. A. and Slaman, T. A., Working below a high recursively enumerable degree, to appear.
Simpson, S. G., First order theory of the degrees of unsolvability, Ann. of Math. (2) 105 (1977), 121–139.Google Scholar
Slaman, T. A., Degree structures, in “Proc. Int. Congress of Math., Kyoto, 1990,” Springer-Verlag, Tokyo, 1991, pp. 303–316.
Slaman, T. A. and Woodin, W. H., Definability in the Turing degrees, Illinois J. Math. 30 (1986), 320–334.Google Scholar
Slaman, T. A. and Woodin, W. H., Definability in degree structures, to appear.
Soare, R. I., “Recursively enumerable sets and degrees,” Springer-Verlag, Berlin, Heidelberg, London, New York, 1987.
Turing, A. M., On computable numbers, with an application to the Entscheidungsproblem, Proc. London Math. Soc. 42 (1936), 230–265.Google Scholar
Turing, A. M., Systems of logic based on ordinals, Proc. London Math. Soc. 45 (1939), 161–228.Google Scholar
Welch, L. V., A Hierarchy of Families of Recursively Enumerable Degrees and a Theorem on Bounding Minimal Pairs, Ph.D. Dissertation, University of Illinois at Urbana-Champaign, 1981.
Yates, C. E. M., Initial segments of the degrees of unsolvability, Part I: A survey, in “Mathematical Logic and Foundations of Set Theory,” (Bar-Hillel, Y., ed.), North-Holland, Amsterdam, 1970, pp. 63–83.
Yates, C. E. M., Initial segments and implications for the structure of degrees, in “Conference in Mathematical Logic, London, 1970,” (Hodges, W., ed.), Lecture Notes in Mathematics No. 255, Springer-Verlag, Berlin, Heidelberg, New York, 1972, pp. 305–335.

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