. There is a comeager set C contained in the set of 1-generic reals and a rst order structure M such that for any real number X, there is an element of C which is recursive in X if and only if there is a presentation of M which is recursive in X. 1. Introduction The theory of a generic real is the theory of the almost everywhere behavior of all of the reals, and as such it can be well approximated. Consequently, constructions which can be implemented relative to any generic real can usually be simulated by approximation. For example, if a set of natural numbers X is recursive in every element of a co-meager set, then X is recursive. Similar statements for arithmetic in or constructible from are equally valid. Counter to these observations, Slaman [1] produced a rst order structure M such that for all reals X, X is not recursive if and only if there is a presentation of M which is recursive in X. X's computing a presentation of M gives an existential criterion for determining whet...

A central concern of computable model theory is the restriction that algebraic structure imposes on the information content of an object of study. One asks about a countable object, what information is coded intrinsically into this object, which cannot be avoided by passing to an isomorphic copy of the object? Given a