On the complexity of models of arithmetic

Journal of Symbolic Logic 47 (2):403-415 (1982)
Abstract
Let P 0 be the subsystem of Peano arithmetic obtained by restricting induction to bounded quantifier formulas. Let M be a countable, nonstandard model of P 0 whose domain we suppose to be the standard integers. Let T be a recursively enumerable extension of Peano arithmetic all of whose existential consequences are satisfied in the standard model. Then there is an initial segment M ' of M which is a model of T such that the complete diagram of M ' is Turing reducible to the atomic diagram of M. Moreover, neither the addition nor the multiplication of M is recursive
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DOI 10.2307/2273150
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Diophantine Induction.Richard Kaye - 1990 - Annals of Pure and Applied Logic 46 (1):1-40.
On Gödel Incompleteness and Finite Combinatorics.Akihiro Kanamori & Kenneth McAloon - 1987 - Annals of Pure and Applied Logic 33 (1):23-41.

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