We consider the set of jumps below a Turing degree, given by , with a focus on the problem: Which recursively enumerable (r.e.) degrees are uniquely determined by ? Initially, this is motivated as a strategy to solve the rigidity problem for the partial order of r.e. degrees. Namely, we show that if every high r.e. degree is determined by , then cannot have a nontrivial automorphism. We then defeat the strategy—at least in the form presented—by constructing pairs , of distinct r.e. degrees such that within any possible jump class . We give some extensions of the construction and suggest ways to salvage the attack on rigidity.
References
[1] Arslanov, M., S. Lempp, and R. A. Shore, “Interpolating d-r.e. and REA degrees between r.e. degrees,” Annals of Pure and Applied Logic, vol. 78 (1996), pp. 29–56.[1] Arslanov, M., S. Lempp, and R. A. Shore, “Interpolating d-r.e. and REA degrees between r.e. degrees,” Annals of Pure and Applied Logic, vol. 78 (1996), pp. 29–56.
[3] Hirschfeldt, D. R., and R. A. Shore, “Combinatorial principles weaker than Ramsey’s theorem for pairs,” Journal of Symbolic Logic, vol. 72 (2007), pp. 171–206.[3] Hirschfeldt, D. R., and R. A. Shore, “Combinatorial principles weaker than Ramsey’s theorem for pairs,” Journal of Symbolic Logic, vol. 72 (2007), pp. 171–206.
[4] Jockusch, C. G., Jr., and R. A. Shore, “Pseudojump operators, I, The r.e. case,” Transactions of the American Mathematical Society, vol. 275 (1983), pp. 599–609.[4] Jockusch, C. G., Jr., and R. A. Shore, “Pseudojump operators, I, The r.e. case,” Transactions of the American Mathematical Society, vol. 275 (1983), pp. 599–609.
[5] Jockusch, C. G., Jr., and R. I. Soare, “Degrees of orderings not isomorphic to recursive linear orderings,” Annals of Pure and Applied Logic, vol. 52 (1991), pp. 39–64.[5] Jockusch, C. G., Jr., and R. I. Soare, “Degrees of orderings not isomorphic to recursive linear orderings,” Annals of Pure and Applied Logic, vol. 52 (1991), pp. 39–64.
[6] Lerman, M., “Automorphism bases for the semilattice of recursively enumerable degrees,” Notices of the American Mathematical Society, vol. 24:A–251, 1977. Abstract no. 77T-E10.[6] Lerman, M., “Automorphism bases for the semilattice of recursively enumerable degrees,” Notices of the American Mathematical Society, vol. 24:A–251, 1977. Abstract no. 77T-E10.
[7] Nies, A., R. A. Shore, and T. A. Slaman, “Interpretability and definability in the recursively enumerable degrees,” Proceedings of the London Mathematical Society, (3), vol. 77 (1998), pp. 241–91.[7] Nies, A., R. A. Shore, and T. A. Slaman, “Interpretability and definability in the recursively enumerable degrees,” Proceedings of the London Mathematical Society, (3), vol. 77 (1998), pp. 241–91.
[10] Shore, R. A., “Some more minimal pairs of $\alpha$-recursively enumerable degrees,” Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 24 (1978), pp. 409–18.[10] Shore, R. A., “Some more minimal pairs of $\alpha$-recursively enumerable degrees,” Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 24 (1978), pp. 409–18.
[12] Soare, R. I., and M. Stob, “Relative recursive enumerability,” pp. 299–324, in Proceedings of the Herbrand Symposium (Marseilles, 1981), vol. 107 of Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1982.[12] Soare, R. I., and M. Stob, “Relative recursive enumerability,” pp. 299–324, in Proceedings of the Herbrand Symposium (Marseilles, 1981), vol. 107 of Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1982.