caa a22 4500 388039930 CHVBK 20180307125018.0 cr unu---uuuuu 161130e199809 xx s 000 0 eng 10.2307/2586727 doi S0022481200014742 pii (NATIONALLICENCE)cambridge-10.2307/2586727 McAllister Alex M. Department of Mathematics, Dartmouth College, Hanover, NH 03755, USA E-mail: Alex.McAllister@dartmouth.edu Completions of PA: Models and enumerations of representable sets [Elektronische Daten] [Alex M. McAllister] We generalize a result on True Arithmetic (ℐA) by Lachlan and Soare to certain other completions of Peano Arithmetic (PA). If ℐ is a completion of PA, then Rep(ℐ) denotes the family of sets X ⊆ ω for which there exists a formula φ(x) such that for all n ∈ ω, if n ∈ X, then ℐ ⊢ φ(S (n) (0)) and if n ∉ X, then ℐ ⊢ ┐φ(S(n)(O)). We show that if S, J ⊆ P(ω) such that S is a Scott set, J is a jump ideal, S ⊂ J and for all X ∈ J, there exists C ∈ S such that C is a "coding” set for the family of subtrees of 2<ω computable in X, and if ℐ is a completion of PA Such that Rep(ℐ) = S, then there exists a model A of ℐ such that J is the Scott set of A and no enumeration of Rep(ℐ) is computable in A. The model A of ℐ is obtained via a new notion of forcing. Before proving our main result, we demonstrate the existence of uncountably many different pairs (S, J) satisfying the conditions of our theorem. This involves a new characterization of 1-generic sets as coding sets for the computable subtrees of 2<ω. In particular, C C ⊆ ω is a coding set for the family of subtrees of 2<ω computable in X if and only if for all trees T ⊆ 2<ω computable in X, if χc is a path through T, then there exists σ ∈ T such that σ ⊂ χc and every extension of σ is in T. Jockusch noted a connection between 1-generic sets and coding sets for computable subtrees of 2<ω. We show they are identical. Copyright © Association for Symbolic Logic 1998 The Journal of Symbolic Logic Cambridge University Press 63/3(1998-09), 1063-1082 0022-4812 63:3<1063 1998 63 JSL https://doi.org/10.2307/2586727 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.2307/2586727 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- McAllister Alex M. Department of Mathematics, Dartmouth College, Hanover, NH 03755, USA E-mail: Alex.McAllister@dartmouth.edu NATIONALLICENCE 773 0- The Journal of Symbolic Logic Cambridge University Press 63/3(1998-09), 1063-1082 0022-4812 63:3<1063 1998 63 JSL CC0 http://creativecommons.org/publicdomain/zero/1.0 nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-cambridge