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A new look at the interpolation problem

Published online by Cambridge University Press:  12 March 2014

Jacques Stern
Jacques Stern*
Affiliation:
UniversitÉ Paris VII, 75005 Paris, France
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Extract

The original aim of this paper was to show that forcing provided a useful and unifying tool in the model theory of finite and admissible languages. Roughly speaking, any result obtained by a Henkin proof, by Makkai's method or by an omitting type argument can be given an alternative proof via forcing. Finally, instead of giving new proofs of a number of known theorems, we have chosen to focus on the interpolation theorem for many-sorted languages. The main result of this paper is thus a generalization of Feferman's theorems ([2], [3]) with a completely different proof; this was announced in [8] and [9].

For more on forcing techniques in model theory the reader should consult [4] as well as [5]. He should also compare forcing and boolean-valued models developed in [6] for the classical case and in [7] for admissible languages. Throughout the paper familiarity with the standard concepts of model theory is assumed.

The author wishes to express his gratitude to J. L. Krivine and K. McAloon for supervising his work. Thanks are also due to J. P. Ressayre for helpful suggestions and to R. L. Vaught for an interesting discussion on forcing in model theory. Finally, an interesting application has been pointed out by S. Feferman and has been included with his permission.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 40 , Issue 1 , March 1975 , pp. 1 - 13
Copyright
Copyright © Association for Symbolic Logic 1975

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References

REFERENCES

[1]Barwise, J., Infinitary languages and admissible sets, Thesis, Stanford, 1967.Google Scholar
[2]Feferman, S., Lectures on proof theory, Proceedings of the summer school, Leeds, 1967, lecture notes, no. 70, pp. 1–107.Google Scholar
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[8]Stern, J., Théorèmes d'interpolation obtenus par la méthode du forcing, Comptes Rendus Hebdomadaires des Séances de l'Académic des Sciences. Séries A et B, vol. 275 (1972).Google Scholar
[9]Stern, J., Un théorème général d'interpolation, this Journal, vol. 39 (1974), p. 384 (abstract).Google Scholar