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Completions of PA: Models and enumerations of representable sets

  • Alex M. McAllister (a1)

Abstract

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 nX, then ℐ ⊢ φ(S(n) (0)) and if nX, then ℐ ⊢ ┐φ(S(n)(O)). We show that if S, JP(ω) such that S is a Scott set, J is a jump ideal, SJ and for all XJ, there exists CS such that C is a “coding” set for the family of subtrees of 2 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. In particular, C C ⊆ ω is a coding set for the family of subtrees of 2 computable in X if and only if for all trees T ⊆ 2 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. We show they are identical.

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References

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Completions of PA: Models and enumerations of representable sets

  • Alex M. McAllister (a1)

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