Relative properties of reals

This paper examines several properties of reals in some relative context. We consider in detail reals which are relatively recursively enumerable, reals which are n-generic relative to some perfect tree, and reals which are relatively hyperimmune-free. We seek to classify which reals hold these properties and study the implications of certain reals being included or excluded.;Many of the findings are unexpected. All but countably many reals are n-generic relative to some perfect tree and relatively hyperimmune-free. However much of the hierarchy of iterated hyperjumps do not hold these properties. Indeed, for genericity we need ZFC-- and infinitely many iterates of the power set of o to complete the proof. The set of relatively recursively enumerable reals is, in some sense, as large as possible. However every nonempty P01 class and the set of relatively REA reals each contain a real which is not relatively recursively enumerable.;We say that a real X is relatively r.e. if there exists a real Y such that X is r.e. (Y ) and X ≰ T Y. We say X is relatively REA if there exists such a Y ≤T X. We define A ≤e 1 B if there exists a Sigma1 set C such that n ∈ A if and only if there is a finite E ⊆ B with ( n, E) ∈ C. We show that a real X is relatively r.e. if and only if X ≰e 1 X..;We prove that every nonempty P01 class contains a real which is not relatively r.e. We also construct a real which is relatively r.e. but not relatively REA. We show that for all reals X and Z such that X ≰e1 X, X