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 $p \lTb \phi$ is equivalent to that of a Boolean-valued model $M$ such that

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

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

References

[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)