# Can the reduced product construction generate boolean-valued models?

In model theory, the reduced product construction contains a collection of structures or models, a set I that indexes the collection, and a filter U on I. Ultraproducts are a special case of reduced products, where U is an ultrafilter.

Boolean-valued models are used primarily in connection with independence proofs in set theory, for example in Rosser's book Simplified Independence Proofs. A boolean-valued model contains a boolean algebra B, and each sentence S of the language L is assigned an element of B as a value. These differ from standard models in which either S or ~S is satisfied at each model. Or one may say the standard models use a 2-valued boolean algebra.

Proposed construction: Take the index set I of a reduced product construction as a set of elements and give it the powerset boolean algebra P(I). Each sentence S of the language L is assigned an element of P(I), namely the set containing the indexes of the models at which it's satisfied. Use the filter U to define the quotient algebra P(I)/U. Then each sentence S of the language has a derived value in the quotient algebra. If the value assigned to S is in the filter U then S's value in the quotient algebra will be the unit element, and the value of ~S will be zero element. If U is an ultrafilter, every S will be asssigned the unit or the zero, but otherwise some sentences will have other values. This generates boolean-valued models for the language L.

This is only an outline; many details are missing. But it appears to show how to use a reduced product to construct a boolean-valued model. Does anyone know if this will or won't work, or has been done already?

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Mike: I'm not sure what you are trying to construct? Are you trying to construct a Boolean-valued model? Do you intend to use this for forcing related constructions, or in general? – Asaf Karagila Aug 4 '11 at 21:46
@Asaf: yes, I'm trying to construct boolean-valued models. The reduced product construction is for first-order logic in general. The boolean-valued models for set theory are special-purpose. I'm thinking if we cross-breed the two we might find something useful, but I'm not sure what exactly. Forcing is not in my cross hairs at the moment ;-). – MikeC Aug 4 '11 at 22:01
Mike: The idea seems reasonable, try searching it on MathSciNet (or Google Scholar if you can't use the former). If you find nothing, well.. this is a good exercise in verifying definitions. Sort of an extension of Los' theorem to Boolean-valued models. – Asaf Karagila Aug 4 '11 at 22:18