Boolean-valued model

A model defined as follows. Let $ \Omega $ be the signature of some first-order language $ L $ with one kind of variables, i.e. $ \Omega $ is the set of symbols of functions and predicates. A Boolean-valued model then is a triple $ M = (B _ {M} , V _ {M} , \Omega _ {M} ) $, where $ B _ {M} $ is a non-degenerate Boolean algebra, $ V _ {M} $ is a non-empty set, and $ \Omega _ {M} $ is a function defined on $ \Omega $ such that

$$ \Omega _ {M} ( \rho ) \in \ V _ {M} ^ {V _ {M} ^ {n} } $$

if $ \rho $ is an $ n $- place function symbol, and

$$ \Omega _ {M} ( \rho ) \in \ B _ {M} ^ {V _ {M} ^ {n} } $$

if $ \rho $ is an $ n $- place predicate symbol. The symbol $ X ^ {Y} $ denotes the set of all functions defined on $ Y $ with values in $ X $ and $ X ^ {n} = X ^ {\{ {m } : {m<n } \} } $, where $ n \geq 0 $ is a natural number. The Boolean algebra $ B _ {M} $ is called the set of truth values of the model $ M $. The set $ V _ {M} $ is called the universe of $ M $. A Boolean-valued model $ M $ is also called a $ B $- model if the set of truth values is the Boolean algebra $ B $, $ B _ {M} = B $. If a Boolean algebra $ B $ is a two-element algebra (i.e. $ B = \{ 0, 1 \} $), then the $ B $- model $ M $ is the classical two-valued model.

Let $ L _ {M} $ be a language, complemented by new individual constants: each $ v \in V _ {M} $ having its own individual constant $ \mathbf v $. Let $ M $ be a $ B $- model and let $ B = (B; 0, 1, C, \cup , \cap ) $ be a complete Boolean algebra; the equalities 1)–8) below then define the value $ \| e \| _ {M} $ of each closed expression $ e $( i.e. of a formula or a term without free variables) of $ L _ {M} $:

1) $ \| \mathbf v \| _ {M} = v $, where $ v \in V _ {M} ; $

2) $ \| \rho ( \tau _ {1} \dots t _ {n} ) \| _ {M} = ( \Omega _ {M} ( \rho )) ( \| \tau _ {1} \| _ {M} \dots \| \tau _ {n} \| _ {M} ), $ where $ \tau _ {1} \dots \tau _ {n} $ are closed terms and $ \rho $ is an $ n $- place function or predicate symbol;

3) $ \| \phi \supset \psi \| _ {M} = - \| \phi \| _ {M} \cup \| \psi \| _ {M} ; $

4) $ \| \phi \lor \psi \| _ {M} = \| \phi \| _ {M} \cup \| \psi \| _ {M} ; $

5) $ \| \phi \wedge \psi \| _ {M} = \| \phi \| _ {M} \cap \| \psi \| _ {M} ; $

6) $ \| \neg \phi \| _ {M} = - \| \phi \| _ {M} ; $

7) $ \| \exists \xi \phi ( \xi ) \| _ {M} = \cup _ {v \in V _ {M} } \| \phi ( \mathbf v ) \| _ {M} ; $

8) $ \| \forall \xi \phi ( \xi ) \| _ {M} = \cap _ {v \in V _ {M} } \| \phi ( \mathbf v ) \| _ {M} . $

The relations 1)–8) define the value $ \| e \| _ {M} $ 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 $ V _ {M} $.

A closed formula $ \phi $ is said to be true in a $ B $- model $ M $( $ M \vDash \phi $) if

$$ \| \phi \| _ {M} = 1. $$

A $ B $- model $ M $ is said to be a model of a theory $ T $ if $ M \vDash \phi $ for all axioms $ \phi $ of $ T $. If $ h $ is a homomorphism of a Boolean algebra $ B $ into a Boolean algebra $ B ^ { \prime } $ preserving infinite unions and intersections, then there exists a $ B ^ { \prime } $ model $ M ^ \prime $ such that

$$ \| \phi \| _ {M ^ \prime } = \ h ( \| \phi \| _ {M} ) $$

for each closed formula $ \phi $ of $ L _ {M} $. If the universe of a model $ M $ is countable, then there exists a homomorphism $ h $ into the Boolean algebra $ \{ 0, 1 \} $, under which $ M $ is transformed into the classical two-valued model $ M ^ \prime $ such that

$$ M \vDash \phi \rightarrow M ^ \prime \vDash \phi . $$

It has been shown that a theory $ T $ 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 $ T $ is constructed by means of another axiomatic theory $ S $, then one obtains the statement on the consistency of $ T $ relative to $ S $. Thus, the result due to P. Cohen on the consistency of the theory $ \mathop{\rm ZF} + (2 ^ {\aleph _ {0} } > \aleph _ {1)} $ 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 $ p \Vdash \phi $ is equivalent to that of a Boolean-valued model $ M $ such that

$$ \| \phi \| _ {M} = \{ {p } : {p \Vdash \neg \neg \phi } \} . $$

References

[1]	E. Rasiowa, R. Sikorski, "The mathematics of metamathematics" , Polska Akad. Nauk (1963)
[2]	T.J. Jech, "Lectures in set theory: with particular emphasis on the method of forcing" , Lect. notes in math. , 217 , Springer (1971)
[3]	G. Takeuti, W.M. Zaring, "Axiomatic set theory" , Springer (1973)
[4]	Yu.I. Manin, "The problem of the continuum" J. Soviet Math. , 5 : 4 (1976) pp. 451–502 Itogi Nauk. i Tekhn. Sovrem. Problemy , 5 (1975) pp. 5–73 Zbl 0375.02055
[a1]	J.L. Bell, "Boolean-valued models and independence proofs in set theory" , Clarendon Press (1977)
[a2]	T.J. Jech, "Set theory" , Acad. Press (1978) (Translated from German)
[a3]	K. Kunen, "Set theory" , North-Holland (1980)