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B. COOPER THEOREM10++ (Slaman and Woodin [SWta]). phism ofT>. Then /(a) = a for all a > 0". Let f be an automor- Other results are proved making full use of Theorem 17 above. Their most dramatic result is: THEOREM 18 (Slaman and Woodin [SWtα]). degrees form an automorphism base for *D. e. degrees) to be rigid. e. >(< 0;) or of D. Increasingly, not only does one find the more intractable and technically interesting questions of degree theory at the local level, but therein is seen to lie the key to the main outstanding problems of global degree theory.

BALDWIN, R. N. McKENZIE, Counting models in universal Horn classes, Algebra universalis, vol. 15 (1982), pp. 359-384. [Bui] S. BUECHLER, The geometry of weakly minimal types, The journal of symbolic logic, vol. 50 (1985), pp. 1044-1054. [Bu2] S. BUECHLER, The classification of small weakly minimal sets /, Classification theory, Chicago 1985, ed. J. Baldwin, Springer 1987, pp. 32-71. [Bu3] S. BUECHLER, Classification of small weakly minimal sets III |, manuscript. [Bu4] S. BUECHLER, Rank 2 types vis-a-vis Vaught's conjecture, preprint 1989.

But: QUESTION 1. Do there exist degrees a, b, a^b, withV(> a) 9*P(> b)? QUESTION 2. ffowfar can Theorems 2+, 7+, 8+ and 10++ be improved? For instance, is any automorphism ofΊ) the identity above 0' ? QUESTION 3. Do there exist non trivia/ automorphisms ofD(< 0') or of Definability. Shore [Sh88] has shown that all levels of the high/low hierarchy from the three level onwards are definable inl>(< 0'), and Shore and Slaman [SStα] have succeeded in distinguishing between the high and low degrees within Ί>(< 0').

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