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A sharp version of the bounded Matijasevich conjecture and the end-extension problem

Published online by Cambridge University Press:  12 March 2014

Zofia Adamowicz
Zofia Adamowicz*
Affiliation:
Institute of Mathematics, Polish Academy of Sciences, 00-950 Warsaw, Poland
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Extract

Under a sharp version of the assumption I∆0 ⊢ ¬ ∆0H we characterize models of I∆0 + BΣ1 having a proper end extension to a model of I∆0.

In [WP] Wilkie and Paris study the relationship between the existence of a proper end extension of a model and its “fullness”, which is related to a certain weak overspill principle (we recall the definition of fullness in §3). Let M be a countable nonstandard model of I∆0 + BΣ1. Under the hypothesis I∆0 ⊢ ¬ ∆0H they prove the following (Corollaries 7 and 8):

The following are equivalent:

  1. 1) M has a proper end extension to a model of I∆0 + BΣ1.

  2. 2) M is (I∆0 + BΣ1)-full.

Moreover, assuming that there is no t ∈ M such that for v ∈ M, 2[t/v]exists if and only if v < N, the following are equivalent:

  1. 1) M has a proper end extension to a model of I∆0.

  2. 2) M is I∆0-full.

Wilkie and Paris ask whether the assumption on the structure of the model can be eliminated from the second equivalence.

We eliminate it, but we sharpen the assumption I∆0 ⊢ ¬ ∆0H. So we partially answer their question.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 57 , Issue 2 , June 1992 , pp. 597 - 616
Copyright
Copyright © Association for Symbolic Logic 1992

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References

REFERENCES

[A]Adamowicz, Z., End extending models of I∆ 0 + exp + BΣ1, Fundamenta Mathematicae, vol. 136 (1990), pp. 133–145.CrossRefGoogle Scholar
[D]Davis, M., Hilbert's tenth problem is unsolvable, American Mathematical Monthly, vol. 80 (1973), pp. 233–269.CrossRefGoogle Scholar
[DG]Dimitracopoulos, C. and Gaifman, H., Fragments of Peano's arithmetic and the MRDP theorem, Logic and algorithmic: an international symposium held in honour of Ernst Specker, Monographies de l'Enseignement Mathématique, no. 30, Université de Genève, Geneva, 1982, pp. 187–206.Google Scholar
[DP]Dimitracopoulos, C. and Paris, J., Truth definitions for ∆ 0 formulae, Logic and algorithmic: an international symposium held in honour of Ernst Specker, Monographies de l'Enseignement Mathématique, no. 30, Université de Genève, Geneva, 1982, pp. 317–329.Google Scholar
[P]Parikh, R., Existence and feasibility in arithmetic, this Journal, vol. 36 (1971), pp. 494–508.Google Scholar
[WP]Wilkie, A. and Paris, J., On the existence of end extensions of models of bounded induction, Logic, methodology and philosophy of science (proceedings of the eighth international congress, Moscow, 1987), North-Holland, Amsterdam, 1989, pp. 143–161.CrossRefGoogle Scholar