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Generic expansions of structures

Published online by Cambridge University Press:  12 March 2014

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

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
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. 119142.Google Scholar
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