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Completions of PA: Models and enumerations of representable sets

Published online by Cambridge University Press:  12 March 2014

Alex M. McAllister
Alex M. McAllister*
Affiliation:
Department of Mathematics, Dartmouth College, Hanover, NH 03755, USA E-mail: [email protected]
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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 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.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 63 , Issue 3 , September 1998 , pp. 1063 - 1082
Copyright
Copyright © Association for Symbolic Logic 1998

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References

REFERENCES

[1]Ash, Chris J. and Knight, Julia F., Computable structures and the hyperarithmetical hierarchy, unpublished manuscript, 1996.Google Scholar
[2]Jockusch, Carl G., Degrees of generic sets, Drake and Wainer, 1980.CrossRefGoogle Scholar
[3]Kaye, Richard, Models of Peano arithmetic, Oxford Logic Guides, no. 15, Oxford University Press, Oxford, New York, 1991.CrossRefGoogle Scholar
[4]Knight, Julia F., Degrees of models with a prescribed Scott set, Proceedings of the U.S.-Israel binational workshop in model theory in mathematical logic: Classification theory, Chicago, 1987, pp. 182–191.Google Scholar
[5]Kumabe, Masahiro, Degrees of generic sets, Computability, enumerability, unsolvability, directions in recursion theory, London Mathematical Society Lecture Note Series, no. 224, 1996, pp. 167–183.CrossRefGoogle Scholar
[6]Lachlan, Alistair H. and Soare, Robert I., Models of arithmetic and upper bounds of arithmetic sets, this Journal, vol. 59 (1994), no. 3, pp. 977–983.Google Scholar
[7]Lerman, Manuel, Degrees of unsolvability, local and global theory, Perspectives in Mathematical Logic, Omega Series, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1980.Google Scholar
[8]Lerman, Manuel, Upper bounds for the arithmetical degrees, Annals of Pure and Applied Logic, vol. 29 (1985), pp. 225–254.CrossRefGoogle Scholar
[9]MacIntyre, Angus and Marker, David, Degrees of recursively saturated models, Transactions of the American Mathematical Society, vol. 282 (1984), no. 2, pp. 539–554.CrossRefGoogle Scholar
[10]Scott, Dana, Algebras of sets binumerable in complete extensions of arithmetic, Recursive function theory, Proceedings of Symposia in Pure Mathematics, no. 5, American Mathematical Society, Providence, 1962, pp. 117–122.CrossRefGoogle Scholar
[11]Scott, Dana and Tennenbaum, Stanley, On the degrees of complete extensions of arithmetic, Notices of the American Mathematical Society, vol. 7 (1960), pp. 242–243, abstract 568-3.Google Scholar
[12]Selman, Alan L., Applications of forcing to the degree-theory of the arithmetical hierarchy, Proceedings of the London Mathematical Society, vol. 25 (1972), pp. 586–602.CrossRefGoogle Scholar
[13]Soare, Robert I., Recursively enumerable sets and degrees, Perspectives in Mathematical Logic, Omega Series, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1987.CrossRefGoogle Scholar
[14]Solovay, Robert M., Degrees of models of true arithmetic, unpublished manuscript, 1982.Google Scholar