Schnorr triviality and genericity

Publisher： Cambridge University Press

E-ISSN： 1943-5886|75|1|191-207

ISSN： 0022-4812

Source： The Journal of Symbolic Logic, Vol.75, Iss.1, 2010-03, pp. : 191-207

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Abstract

We study the connection between Schnorr triviality and genericity. We show that while no 2-generic is Turing equivalent to a Schnorr trivial and no 1-generic is tt-equivalent to a Schnorr trivial, there is a 1-generic that is Turing equivalent to a Schnorr trivial. However, every such 1-generic must be high. As a corollary, we prove that not all K-trivials are Schnorr trivial. We also use these techniques to extend a previous result and show that the bases of cones of Schnorr trivial Turing degrees are precisely those whose jumps are at least 0″.

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