A model defined as follows. Let be the signature of some first-order language with one kind of variables, i.e. is the set of symbols of functions and predicates. A Boolean-valued model then is a triple , where is a non-degenerate Boolean algebra, is a non-empty set, and is a function defined on such that
if is an -place function symbol, and
if is an -place predicate symbol. The symbol denotes the set of all functions defined on with values in and , where is a natural number. The Boolean algebra is called the set of truth values of the model . The set is called the universe of . A Boolean-valued model is also called a -model if the set of truth values is the Boolean algebra , . If a Boolean algebra is a two-element algebra (i.e. ), then the -model is the classical two-valued model.
Let be a language, complemented by new individual constants: each having its own individual constant . Let be a -model and let be a complete Boolean algebra; the equalities 1)–8) below then define the value of each closed expression (i.e. of a formula or a term without free variables) of :
1) , where
2) where are closed terms and is an -place function or predicate symbol;
The relations 1)–8) define the value for certain non-complete Boolean algebras as well; the only condition is that the infinite unions and intersections in 7) and 8) exist. The concept of a Boolean-valued model can also be introduced for languages with more than one kind of variables. In such a case each kind of variable has its own domain of variation .
A closed formula is said to be true in a -model () if
A -model is said to be a model of a theory if for all axioms of . If is a homomorphism of a Boolean algebra into a Boolean algebra preserving infinite unions and intersections, then there exists a model such that
for each closed formula of . If the universe of a model is countable, then there exists a homomorphism into the Boolean algebra , under which is transformed into the classical two-valued model such that
It has been shown that a theory is consistent if and only if it has a Boolean-valued model. This theorem forms the basis of the application of the theory of Boolean-valued models to problems of the consistency of axiomatic theories.
If the Boolean-valued model of a theory is constructed by means of another axiomatic theory , then one obtains the statement on the consistency of relative to . Thus, the result due to P. Cohen on the consistency of the theory relative to ZF is obtained by constructing the respective Boolean-valued model by means of the system ZF (cf. Forcing method). The construction of the Cohen forcing relation is equivalent to that of a Boolean-valued model such that
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Boolean-valued model. V.N. Grishin (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Boolean-valued_model&oldid=17991