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One more aspect of forcing and omitting types

Published online by Cambridge University Press:  12 March 2014

Extract

As has been shown by the investigations of some mathematicians, there are numerous analogies between forcing and omitting types. Evidence of that can also be found in the application of forcing to model theory. Both methods use the Rasiowa-Sikorski lemma on the existence in a Boolean algebra of an ultrafilter intersecting every dense set from a given denumerable family.

In this paper we will not use the terms Cohen forcing and omitting types in their original sense. We shall deal mainly with the Scott-Boolean kind of forcing as a transformation of Cohen's idea and with methods used in logic that consist in finding a suitable ultrafilter in the Lindenbaum-Tarski algebra for a given theory and defining canonical models—under the name of omitting types.

The two methods will be confronted to show that Cohen's forcing stems from logical methods. Attention will also be drawn to some differences between forcing in set theory and the general methods of logic.

In logic we construct a model by defining relations in a set of constants. In particular when defining a model for set theory we define a certain relation E in a set of constants. It is usually immaterial what E is, in particular whether it is the true relation of membership up to isomorphism.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1976

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References

REFERENCES

[1] Mostowski, A., Constructible sets with applications, Studies in logic and the foundations of mathematics, North-Holland, Amsterdam.Google Scholar
[2] Scott, Dana, Boolean-valued models.Google Scholar