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On models of arithmetic—Answers to two problems raised by H. Gaifman

Published online by Cambridge University Press:  12 March 2014

Alex Wilkie
Alex Wilkie*
Affiliation:
University of Leicester, Leicester, England
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Extract

In a recent paper [3] H. Gaifman investigated some model theoretic consequences of Matijasevič's theorem [5], and posed some further problems which naturally arise. We provide here partial answers to two of these problems, the results having been previously announced in the postscript of [3].

Firstly, it is shown in [3] that if M1 and M2 are models of the Peano axioms P and M1 ⊆ M2, then M1 is closed under the recursive functions of M2. The converse of this statement is false. Moreover, Gaifman asks: Is every initial segment of a model M of P which is closed under the recursive functions of M (or the ∑n-definable functions) also a model of P? We show that this is false and our method gives, en route, another proof of a theorem of Rabin [7] stating the P is not implied by any consistent set of ∑n sentences for any n.

Secondly, we partially answer a question posed on p. 129 of [3] by proving (some-what more than) every countable nonstandard model of P has an end extension in which a diophantine equation, not solvable in the original model, has a solution. We can, in fact, take the new model to be isomorphic to the original one. This generalises (apart from the countability restriction) a theorem of Rabin [6].

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 40 , Issue 1 , March 1975 , pp. 41 - 47
Copyright
Copyright © Association for Symbolic Logic 1975

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References

REFERENCES

[1]Friedman, H., Countable models of set theories, Cambridge Summer School in Mathematical Logic, Springer-Verlag Lecture Notes in Mathematics, no. 337, 1973, pp. 539–573.Google Scholar
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