THE TURING DEGREES BELOW GENERICS AND RANDOMS

Abstract

AbstractIf X0and X1are both generic, the theories of the degrees below X0and X1are the same. The same is true if both are random. We show that then-genericity orn-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.