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Arithmetical independence results using higher recursion theory

Published online by Cambridge University Press:  12 March 2014

Andrew Arana
Affiliation:
Department of Philosophy, Building 90, Stanford University, Stanford, CA 94305-2155, USA, E-mail: aarana@stanford.edu, URL: http://www.stanford.edu/~aarana
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Abstract

We extend an independence result proved in [1]. We show that for all n, there is a special set of Πn sentences {φa}a ∈ H corresponding to elements of a linear ordering (H, <H) of order type . These sentences allow us to build completions {Ta}a ∈ H of PA such that for a <H b, Ta ∩ ΣnTb ∩ Σn, with φaTa, ¬φaTh. Our method uses the Barwise-Kreisel Compactness Theorem.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2004

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References

[1]Arana, Andrew, Solovay's theorem cannot be simplified, Annals of Pure and Applied Logic, vol. 112 (2001), no. 1, pp. 2741.CrossRefGoogle Scholar
[2]Ash, Christopher J. and Knight, Julia F., Computable structures and the hyperarithmetical hierarchy, Studies in Logic and the Foundations of Mathematics, vol. 144, Elsevier, Amsterdam, 2000.Google Scholar
[3]D'Aquino, Paola and Knight, Julia F., Coding in IΔ0, 2002, Preprint.Google Scholar
[4]Feferman, Solomon, Arithmetically definable models of formalizable arithmetic, Notices of the American Mathematical Society, vol. 5 (1958), p. 679.Google Scholar
[5]Kaye, Richard, Models of Peano Arithmetic, Oxford Logic Guides, vol. 15, Oxford University Press, Oxford, 1991.Google Scholar
[6]Rogers, Hartley, Theory of recursive functions and effective computability, MIT Press, Cambridge, 1967.Google Scholar
[7]Simmons, Harold, Large discrete parts of the E-tree, this Journal, vol. 53 (1988), no. 3, pp. 980984.Google Scholar

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