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ω 1-like recursively saturated models of Presburger's arithmetic

Published online by Cambridge University Press:  12 March 2014

Victor Harnik
Victor Harnik*
Affiliation:
Department of Mathematics, University of Haifa, Haifa, Israel
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Extract

Let Pr be Presburger's arithmetic, i.e., the complete theory of the structure (ω, +). Lipshitz and Nadel showed in [4] that a countable model of Pr is recursively saturated iff it can be expanded to a model of Peano arithmetic, PA. As a starting point for an introductory discussion, let us mention one more fact about countable recursively saturated models of Pr. If is such a model then the following is readily seen (as explained also in §1):

Any two countable recursively saturated elementary endextensions of are isomorphic.

If we drop “countable” from the assumption of this statement, we can still say that the two models are ∞ω-equivalent. Must they be isomorphic if both have cardinality ℵ1? Certainly not, since one of the models can be ω 1-like while the other is not. Once we realized this much, let us concentrate on ω 1-like structures. We prove in §3:

Theorem. Any countable recursively saturated model of Pr has isomorphism types of ω 1-like recursively saturated elementary endextensions. Only one of these is the isomorphism type of a structure that can be expanded to a model of PA.

The key technical result is proven in §2. It says that an as above has precisely two countable recursively saturated elementary endextensions which are nonisomorphic over .

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 51 , Issue 2 , June 1986 , pp. 421 - 429
Copyright
Copyright © Association for Symbolic Logic 1986

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References

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