Minimal covers and arithmetical sets

Abstract

If a a and b b are degrees of unsolvability, a a is called a minimal cover of b b if b > a b > a and no degree c c satisfies b > c > a b > c > a . The degree a a is called a minimal cover if it is a minimal cover of some degree b b . We prove by a very simple argument that 0 n {0^n} is not a minimal cover for any n n . From this result and the axiom of Borel determinateness (BD) we show that the degrees of arithmetical sets (with their usual ordering) are not elementarily equivalent to all the degrees. We also point out how this latter result can be proved without BD when the jump operation is added to the structures involved.