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THE TURING DEGREES BELOW GENERICS AND RANDOMS

Published online by Cambridge University Press:  17 April 2014

RICHARD A. SHORE*
Affiliation:
DEPARTMENT OF MATHEMATICS, CORNELL UNIVERSITY, ITHACA NY 14853, USAE-mail:[email protected]

Abstract

If X0 and X1 are both generic, the theories of the degrees below X0 and X1 are the same. The same is true if both are random. We show that the n-genericity or n-randomness of X do not suffice to guarantee that the degrees below X have these common theories. We also show that these two theories (for generics and randoms) are different. These results answer questions of Jockusch as well as Barmpalias, Day and Lewis.

Type
Articles
Copyright
Copyright © Association for Symbolic Logic 2014 

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