Undecidability and 1-types in the recursively enumerable degrees - Ambos-Spies, Klaus; Shore, Richard A. (Annals of pure and applied logic, 1993)
Ambos-Spies, K. and R.A. Shore, Undecidability and 1-types in the recursively enumerable degrees, Annals of Pure and Applied Logic 63 (1993) 3-37. We show that the theory of the partial ordering of recursively enumerable Turing degrees is undecidable and has uncountably many 1-types. In contrast to the original proof of the former which used a very complicated O''' argument our proof proceeds by a much simpler infinite injury argument. Moreover, it combines with the permitting technique to get similar results for any ideal of the r.e. degrees.
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