Computability in structures representing a Scott set

Abstract

Continuing work begun in [10], we utilize a notion of forcing for which the generic objects are structures and which allows us to determine whether these "generic" structures compute certain sets and enumerations. The forcing conditions are bounded complexity types which are consistent with a given theory and are elements of a given Scott set. These generic structures will "represent" this given Scott set, in the sense that the structure has a certain weak saturation property with respect to bounded complexity types in the Scott set. For example, if Script A sign is a nonstandard model of PA, then A-fraktur sign represents the Scott set Script L sign = {{n ∈ | Script A sign |= "the nth prime divides a"} | a ∈ Script A sign}. The notion of forcing yields two main results. The first characterizes the sets of natural numbers computable in all models of a given theory representing a given Scott set. We show that the characteristic function of such a set must be enumeration reducible to a complete existential type which is consistent with the given theory and is an element of the given Scott set. The second provides a sufficient condition for the existence of a structure Script A sign such that Script A sign represents a countable jump ideal and Script A sign does not compute an enumeration of a given family of sets Script L sign. This second result is of particular interest when the family of sets which cannot be enumerated is Script L sign = Rep[Th(Script A sign)]. Under this additional assumption, the second result generalizes a result on TA [6] and on certain other completions of PA [10]. For example, we show that there also exist models of completions of ZF from which one cannot enumerate the family of sets represented by the theory.