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A set $G \subseteq \omega$ is n-generic for a positive integer n if and only if every ${\rm{\Sigma }}_n^0$ formula of G is decided by a finite initial segment of G in the sense of Cohen forcing. It is shown here that every n-generic set G is properly ${\rm{\Sigma }}_n^0$ in some G-recursive X. As a corollary, we also prove that for every $n > 1$ and every n-generic set G there exists a G-recursive X which is generalized ${\rm{lo}}{{\rm{w}}_n}$ but not generalized ${\rm{lo}}{{\rm{w}}_{n - 1}}$ . Thus we confirm two conjectures of Jockusch [4].



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[1]Anderson, B. A., Relatively computably enumerable reals. Archive for Mathematical Logic, vol. 50 (2011), no. 3–4, pp. 361365.
[2]Cai, M. and Shore, R., Domination, forcing, array nonrecursiveness and relative recursive enumerability, this Journal, vol. 77 (2012), pp. 226239.
[3]Downey, R. and Hirschfeldt, D., Algorithmic Randomness and Complexity, Theory and Applications of Computability, Springer, 2010.
[4]Jockusch, C. G. Jr., Degrees of generic sets, Recursion Theory: Its Generalizations and Applications, Proceedings of Logic Colloquium ’79 (Drake, F. R. and Wainer, S. S., editors), Cambridge University Press, Cambridge, 1980, pp. 110139.
[5]Kumabe, M., Relative recursive enumerability of generic degrees, this Journal, vol. 56 (1991), no. 3, pp. 10751084.
[6]Kumabe, M., Degrees of generic sets, Computability, Enumerability, Unsolvability (Cooper, S. B., Slaman, T. A., and Wainer, S. S., editors), London Mathematical Society, Lecture Notes Series, vol. 224, Cambridge University Press, Cambridge, 1996, pp. 167183.
[7]Kurtz, S., Randomness and genericty in the degrees of unsolvability, Ph.D. thesis, University of Illinios at Urbana-Champaign, 1981.
[8]Yates, C. E. M., Initial segments of the degrees of unsolvability, part II: Minimal degrees, this Journal, vol. 35 (1970), pp. 243266.



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