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A model defined as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b0169901.png" /> be the signature of some first-order language <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b0169902.png" /> with one kind of variables, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b0169903.png" /> is the set of symbols of functions and predicates. A Boolean-valued model then is a triple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b0169904.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b0169905.png" /> is a non-degenerate [[Boolean algebra|Boolean algebra]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b0169906.png" /> is a non-empty set, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b0169907.png" /> is a function defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b0169908.png" /> such that
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b0169909.png" /></td> </tr></table>
+
{{TEX|auto}}
 +
{{TEX|done}}
  
if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b01699010.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b01699011.png" />-place function symbol, and
+
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|Boolean algebra]],  $  V _ {M} $
 +
is a non-empty set, and $  \Omega _ {M} $
 +
is a function defined on  $  \Omega $
 +
such that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b01699012.png" /></td> </tr></table>
+
$$
 +
\Omega _ {M} ( \rho )  \in \
 +
V _ {M} ^ {V _ {M}  ^ {n} }
 +
$$
  
if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b01699013.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b01699014.png" />-place predicate symbol. The symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b01699015.png" /> denotes the set of all functions defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b01699016.png" /> with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b01699017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b01699018.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b01699019.png" /> is a natural number. The Boolean algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b01699020.png" /> is called the set of truth values of the model <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b01699021.png" />. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b01699022.png" /> is called the universe of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b01699023.png" />. A Boolean-valued model <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b01699024.png" /> is also called a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b01699026.png" />-model if the set of truth values is the Boolean algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b01699027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b01699028.png" />. If a Boolean algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b01699029.png" /> is a two-element algebra (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b01699030.png" />), then the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b01699031.png" />-model <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b01699032.png" /> is the classical two-valued model.
+
if $  \rho $
 +
is an $  n $-
 +
place function symbol, and
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b01699033.png" /> be a language, complemented by new individual constants: each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b01699034.png" /> having its own individual constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b01699035.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b01699036.png" /> be a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b01699037.png" />-model and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b01699038.png" /> be a complete Boolean algebra; the equalities 1)–8) below then define the value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b01699039.png" /> of each closed expression <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b01699040.png" /> (i.e. of a formula or a term without free variables) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b01699041.png" />:
+
$$
 +
\Omega _ {M} ( \rho ) \in \
 +
B _ {M} ^ {V _ {M}  ^ {n} }
 +
$$
  
1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b01699042.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b01699043.png" />
+
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.
  
2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b01699044.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b01699045.png" /> are closed terms and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b01699046.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b01699047.png" />-place function or predicate symbol;
+
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} $:
  
3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b01699048.png" />
+
1) $  \| \mathbf v \| _ {M} = v $,
 +
where  $  v \in V _ {M} ; $
  
4) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b01699049.png" />
+
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;
  
5) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b01699050.png" />
+
3) $  \| \phi \supset \psi \| _ {M} = - \| \phi \| _ {M} \cup \| \psi \| _ {M} ; $
  
6) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b01699051.png" />
+
4) $  \| \phi \lor \psi \| _ {M} = \| \phi \| _ {M} \cup \| \psi \| _ {M} ; $
  
7) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b01699052.png" />
+
5) $  \| \phi \wedge \psi \| _ {M} = \| \phi \| _ {M} \cap \| \psi \| _ {M} ; $
  
8) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b01699053.png" />
+
6) $  \| \neg \phi \| _ {M} = - \| \phi \| _ {M} ; $
  
The relations 1)–8) define the value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b01699054.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b01699055.png" />.
+
7) $  \| \exists \xi  \phi ( \xi ) \| _ {M} = \cup _ {v \in V _ {M}  } \| \phi ( \mathbf v ) \| _ {M} ; $
  
A closed formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b01699056.png" /> is said to be true in a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b01699057.png" />-model <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b01699058.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b01699059.png" />) if
+
8)  $  \| \forall \xi  \phi ( \xi ) \| _ {M} = \cap _ {v \in V _ {M}  } \| \phi ( \mathbf v ) \| _ {M} . $
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b01699060.png" /></td> </tr></table>
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b01699061.png" />-model <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b01699062.png" /> is said to be a model of a theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b01699063.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b01699064.png" /> for all axioms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b01699065.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b01699066.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b01699067.png" /> is a homomorphism of a Boolean algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b01699068.png" /> into a Boolean algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b01699069.png" /> preserving infinite unions and intersections, then there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b01699070.png" /> model <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b01699071.png" /> such that
+
A closed formula  $  \phi $
 +
is said to be true in a $  B $-
 +
model $  M $(
 +
$  M \vDash \phi $)
 +
if
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b01699072.png" /></td> </tr></table>
+
$$
 +
\| \phi \| _ {M}  = 1.
 +
$$
  
for each closed formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b01699073.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b01699074.png" />. If the universe of a model <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b01699075.png" /> is countable, then there exists a homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b01699076.png" /> into the Boolean algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b01699077.png" />, under which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b01699078.png" /> is transformed into the classical two-valued model <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b01699079.png" /> such that
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b01699080.png" /></td> </tr></table>
+
$$
 +
\| \phi \| _ {M  ^  \prime  }  = \
 +
h ( \| \phi \| _ {M} )
 +
$$
  
It has been shown that a theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b01699081.png" /> 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.
+
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
  
If the Boolean-valued model of a theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b01699082.png" /> is constructed by means of another axiomatic theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b01699083.png" />, then one obtains the statement on the consistency of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b01699084.png" /> relative to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b01699085.png" />. Thus, the result due to P. Cohen on the consistency of the theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b01699086.png" /> relative to ZF is obtained by constructing the respective Boolean-valued model by means of the system ZF (cf. [[Forcing method|Forcing method]]). The construction of the Cohen forcing relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b01699087.png" /> is equivalent to that of a Boolean-valued model <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b01699088.png" /> such that
+
$$
 +
M \vDash \phi  \rightarrow  M  ^  \prime  \vDash \phi .
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016990/b01699089.png" /></td> </tr></table>
+
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|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====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Rasiowa,  R. Sikorski,  "The mathematics of metamathematics" , Polska Akad. Nauk  (1963)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  T.J. Jech,  "Lectures in set theory: with particular emphasis on the method of forcing" , ''Lect. notes in math.'' , '''217''' , Springer  (1971)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  G. Takeuti,  W.M. Zaring,  "Axiomatic set theory" , Springer  (1973)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  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</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Rasiowa,  R. Sikorski,  "The mathematics of metamathematics" , Polska Akad. Nauk  (1963)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  T.J. Jech,  "Lectures in set theory: with particular emphasis on the method of forcing" , ''Lect. notes in math.'' , '''217''' , Springer  (1971)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  G. Takeuti,  W.M. Zaring,  "Axiomatic set theory" , Springer  (1973)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  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</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.L. Bell,  "Boolean-valued models and independence proofs in set theory" , Clarendon Press  (1977)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  T.J. Jech,  "Set theory" , Acad. Press  (1978)  (Translated from German)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  K. Kunen,  "Set theory" , North-Holland  (1980)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.L. Bell,  "Boolean-valued models and independence proofs in set theory" , Clarendon Press  (1977)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  T.J. Jech,  "Set theory" , Acad. Press  (1978)  (Translated from German)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  K. Kunen,  "Set theory" , North-Holland  (1980)</TD></TR></table>

Latest revision as of 20:15, 29 May 2020


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

Comments

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)
How to Cite This Entry:
Boolean-valued model. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Boolean-valued_model&oldid=17991
This article was adapted from an original article by V.N. Grishin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article