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Generic expansions of structures

Published online by Cambridge University Press:  12 March 2014

Julia F. Knight
Julia F. Knight*
Affiliation:
Pennsylvania State University, University Park, Pennsylvania 16802
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Extract

In this paper, Cohen's forcing technique is applied to some problems in model theory. Forcing has been used as a model-theoretic technique by several people, in particular, by A. Robinson in a series of papers [1], [10], [11]. Here forcing will be used to expand a family of structures in such a way that weak second-order embeddings are preserved. The forcing situation resembles that in Solovay's proof that for any theorem φ of GB (Godel-Bernays set theory with a strong form of the axiom of choice), if φ does not mention classes, then it is already a theorem of ZFC. (See [3, p. 105] and [2, p. 77].)

The first application of forcing here is to the problem (posed by Keisler) of when is it possible to add a Skolem function to a pair of structures, one of which is an elementary substructure of the other, in such a way that the elementary embedding is preserved.

It is not always possible to find such a Skolem function. Payne [9] found an example involving countable structures with uncountably many relations. The author [4], [6] found an example involving uncountable structures with only two relations. The problem remains open in case the structures are required both to be countable and to have countable type. Forcing is used to obtain a positive result under some special conditions.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 38 , Issue 4 , December 1973 , pp. 561 - 570
Copyright
Copyright © Association for Symbolic Logic 1973

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References

REFERENCES

[1] Barwise, J. and Robinson, A., Completing theories by forcing, Annals of Mathematical Logic, vol. 2 (1970), pp. 119–142.Google Scholar
[2] Cohen, P. J., Set theory and the continuum hypothesis, Benjamin, New York and Amsterdam, 1966.Google Scholar
[3] Keisler, H. J., Model theory for infinitary logic, North-Holland, Amsterdam, 1971.Google Scholar
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[6] Knight, J., Some problems in model theory, Doctoral dissertation, University of California, Berkeley, 1972.Google Scholar
[7] Knight, J., Theories with finitely many ω-models, Notices of the American Mathematical Society, vol. 20 (1973), p. A–283.Google Scholar
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[11] Robinson, A., Forcing in model theory, Proceedings of the 1969 Colloquium on Model Theory In Rome (to appear).Google Scholar