COMPLETIONS OF PA - MODELS AND ENUMERATIONS OF REPRESENTABLE SETS

We generalize a result on True Arithmetic (TA) by Lachlan and Soare to certain other completions of Peano Arithmetic (PA). If T is a complet ion of PA, then Rep(T) denotes the family of sets X subset of or equal to omega for which there exists a formula phi(x) such that for all n is an element of omega, if n is an element of X, then T proves phi(S-( n)(0)) and if n is not an element of X, then T proves inverted left pe rpendicular phi(S-(n)(0)). We show that if S, F subset of or equal to P(omega) such that S is a Scott set, F is a jump ideal, S subset of F and for all X is an element of F, there exists C is an element of S su ch that C is a ''coding'' set for the family of subtrees of 2(<omega) computable in X, and if F is a completion of PA such that Rep(T) = S, then there exists a model A of T such that F is the Scott set of A and no enumeration of Rep(T) is computable in A. The model A of T is obta ined via a new notion of forcing. Before proving our main result, we d emonstrate the existence of uncountably many different pairs (S, F) sa tisfying the conditions of our theorem. This involves a new characteri zation of 1-generic sets as coding sets for the computable subtrees of 2(<omega). In particular, C subset of or equal to omega is a coding s et for the family of subtrees of 2(<omega) computable in X if and only if for all trees T subset of or equal to 2(<omega) computable in X, i f chi(C) is a path through T, then there exists sigma is an element of T such that sigma subset of chi(C) and every extension of sigma is in T. Jockusch noted a connection between 1-generic sets and coding sets for computable subtrees of 2(<omega). We show they are identical.