View source for Forcing method

A special method for constructing models of [[Axiomatic set theory|axiomatic set theory]]. It was proposed by P.J. Cohen in 1963 to prove the compatibility of the negation of the [[Continuum hypothesis|continuum hypothesis]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f0407701.png" />, and other set-theoretic assumptions with the axioms of the Zermelo–Fraenkel system ZF [[#References|[1]]]. The forcing method was subsequently simplified and modernized [[#References|[2]]]–[[#References|[6]]]. In particular, it was found that the method was related to the theory of Boolean-valued models (cf. [[Boolean-valued model|Boolean-valued model]]) [[#References|[2]]], [[#References|[3]]] and [[Kripke models|Kripke models]] [[#References|[6]]].

The central concept of the forcing method is the forcing relation

<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/f/f040/f040770/f0407702.png" /></td> </tr></table>

( "the condition p forces the formula f" ).

The definition of the forcing relation is preceded by the specification of a language <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f0407703.png" /> and a partially ordered set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f0407704.png" /> of forcing conditions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f0407705.png" /> with order relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f0407706.png" />. The language <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f0407707.png" /> may contain variables and constants of different sorts (or types).

The construction of the model of ZF proposed by Cohen, in which the continuum hypothesis is violated, proceeds as follows. A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f0407708.png" /> is called transitive 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/f/f040/f040770/f0407709.png" /></td> </tr></table>

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f04077010.png" /> be a denumerable transitive set which is a model of ZF, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f04077011.png" /> be an ordinal number (in the sense of von Neumann), i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f04077012.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f04077013.png" /> be an arbitrary set (possibly <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f04077014.png" />), where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f04077015.png" /> is the first infinite ordinal number. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f04077016.png" /> is a transitive set, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f04077017.png" /> denote the set of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f04077018.png" />-definable subsets (cf. [[Gödel constructive set|Gödel constructive set]]), i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f04077019.png" />. A process similar to that of constructing Gödel constructive sets is used to inductively define a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f04077020.png" /> for any ordinal number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f04077021.png" />,

<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/f/f040/f040770/f04077022.png" /></td> </tr></table>

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f04077023.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f04077024.png" />. The model of ZF in which the continuum hypothesis is violated is to be found among the models of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f04077025.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f04077026.png" /> be an ordinal number such that the statement  "l is the second uncountable ordinal"  is true in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f04077027.png" />.

The set of forcing conditions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f04077028.png" /> and the relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f04077029.png" /> is defined by the following equivalences: a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f04077030.png" /> is a function defined on some finite subset of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f04077031.png" /> with values in the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f04077032.png" />; b) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f04077033.png" /> is an extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f04077034.png" />. The language <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f04077035.png" /> which is taken is a so-called ramified language, with many types of variables (each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f04077036.png" /> having its own type of variables running through the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f04077037.png" />) and containing names (i.e. individual constants) for each set from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f04077038.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f04077039.png" />, the name of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f04077040.png" /> is written as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f04077041.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f04077042.png" /> be the name of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f04077043.png" />. The forcing relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f04077044.png" /> is introduced by an inductive definition which has, in particular, the following characteristics

1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f04077045.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f04077046.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f04077047.png" />;

2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f04077048.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f04077049.png" />;

3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f04077050.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f04077051.png" />;

4) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f04077052.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f04077053.png" />;

if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f04077054.png" /> is the type of the variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f04077055.png" />, then

5) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f04077056.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f04077057.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f04077058.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f04077059.png" /> is the set of all constants of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f04077060.png" />.

A sequence of forcing conditions

<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/f/f040/f040770/f04077061.png" /></td> </tr></table>

is called complete if for any closed formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f04077062.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f04077063.png" /> one has

<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/f/f040/f040770/f04077064.png" /></td> </tr></table>

The countability of the set of all closed formulas of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f04077065.png" /> and 2) above make it possible to demonstrate the existence of a complete sequence, beginning from an arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f04077066.png" />.

A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f04077067.png" /> contained in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f04077068.png" /> is called generic with respect to the model <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f04077069.png" /> if there exists a complete sequence such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f04077070.png" /> is the characteristic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f04077071.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f04077072.png" />. The following two facts about generic sets and the forcing relation are of fundamental importance.

I) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f04077073.png" /> is a generic set, then

<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/f/f040/f040770/f04077074.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f04077075.png" /> means that formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f04077076.png" /> is true in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f04077077.png" />.

II) The relation

<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/f/f040/f040770/f04077078.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f04077079.png" /> are constants in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f04077080.png" />, regarded as a relation between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f04077081.png" />, can be expressed in the model <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f04077082.png" />.

In view of the above facts, it is sufficient to show that the statement

<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/f/f040/f040770/f04077083.png" /></td> </tr></table>

is true in the model <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f04077084.png" />, i.e. to prove that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f04077085.png" />. The verification of the validity of the axioms of ZF and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f04077086.png" /> in the model <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f04077087.png" /> is based on this proposition. The verification of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f04077088.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f04077089.png" /> also involves the use of the special properties of the set of forcing conditions, which makes it possible to show that:

(1)) If the ordinal numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f04077090.png" /> are unequal, one has

<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/f/f040/f040770/f04077091.png" /></td> </tr></table>

that is,

<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/f/f040/f040770/f04077092.png" /></td> </tr></table>

(2)) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f04077093.png" />.

There is the following relation between forcing and Boolean-valued models.

If one introduces the notation

<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/f/f040/f040770/f04077094.png" /></td> </tr></table>

<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/f/f040/f040770/f04077095.png" /></td> </tr></table>

then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f04077096.png" /> is a complete [[Boolean algebra|Boolean algebra]], and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f04077097.png" /> is the Boolean value of the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f04077098.png" />. Thus, to specify a partially ordered set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f04077099.png" /> and to define the relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f040770100.png" /> is equivalent to construct some Boolean-valued model <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f040770101.png" />. The analysis of the proof of theorems of the type:

<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/f/f040/f040770/f040770102.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f040770103.png" /> is an axiom of ZF or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f040770104.png" />, leads to the conclusion that the formulas of ZF expressing the theorem

<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/f/f040/f040770/f040770105.png" /></td> </tr></table>

i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f040770106.png" />, are deducible from the axioms of ZF. Thus, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f040770107.png" /> is a Boolean-valued model for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f040770108.png" />, constructed with the tools of ZF. The assumption of the existence of a denumerable transitive set which is a model of ZF, and the concept of a generic set, are both unessential in the proof of the relative consistency.

It has been shown that the construction of the Boolean-valued model may be simplified [[#References|[2]]], [[#References|[3]]], [[#References|[5]]]. In particular, the introduction of the ramified language <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f040770109.png" /> is not necessary. A generic model <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f040770110.png" /> may also be constructed as follows [[#References|[4]]].

A subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f040770111.png" /> of a partially ordered set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f040770112.png" /> is called dense 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/f/f040/f040770/f040770113.png" /></td> </tr></table>

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f040770114.png" /> and the relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f040770115.png" /> be elements of some denumerable transitive set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f040770116.png" /> which is a model of ZF. A subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f040770117.png" /> is called an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f040770119.png" />-generic filter if

(1)) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f040770120.png" />;

(2)) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f040770121.png" />;

(3)) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f040770122.png" />.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f040770123.png" /> be an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f040770124.png" />-generic filter in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f040770125.png" />. Since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f040770126.png" /> is denumerable, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f040770127.png" /> exists. In general, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f040770128.png" />. The relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f040770129.png" /> is defined by the equivalence

<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/f/f040/f040770/f040770130.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f040770131.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f040770132.png" /> are arbitrary elements of the model <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f040770133.png" />.

Let a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f040770134.png" /> be defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f040770135.png" /> by the equations

<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/f/f040/f040770/f040770136.png" /></td> </tr></table>

and let

<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/f/f040/f040770/f040770137.png" /></td> </tr></table>

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f040770138.png" /> is a closed formula of the language ZF supplemented by constants to denote each set from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f040770139.png" />, one defines

<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/f/f040/f040770/f040770140.png" /></td> </tr></table>

It can be shown that

(I)) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f040770141.png" />.

(II)) The relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f040770142.png" /> can be defined in the model <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f040770143.png" /> for each formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f040770144.png" />.

By using solely (I) and (II) and basing oneself on the fact that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f040770145.png" /> is a model of ZF, it is possible to establish that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f040770146.png" /> is a model of ZF. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f040770147.png" /> is defined by the equivalences a) and b), one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f040770148.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f040770149.png" /> is the characteristic function of some set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f040770150.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f040770151.png" />. The relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f040770152.png" />, which is definable in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f040770153.png" />, does not satisfy points 3) and 5) of Cohen's definition of the forcing relation. One has

<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/f/f040/f040770/f040770154.png" /></td> </tr></table>

If one assumes 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/f/f040/f040770/f040770155.png" /></td> </tr></table>

a Boolean-valued model for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f040770156.png" /> is obtained which is definable in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f040770157.png" /> and has a Boolean algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f040770158.png" /> which is identical with that of Cohen.

Thus, the forcing method consists in fact of constructing a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f040770159.png" />-model and a homomorphism which preserves certain infinite unions and intersections of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f040770160.png" /> into the two-element algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040770/f040770161.png" />. For applications of the forcing method in set theory see, for example, [[#References|[2]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.J. Cohen,   "Set theory and the continuum hypothesis" , Benjamin  (1966)</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">  J.R. Shoenfield,   "Mathematical logic" , Addison-Wesley  (1967)</TD></TR><TR><TD valign="top">[5]</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><TR><TD valign="top">[6]</TD> <TD valign="top">  M.C. Fitting,   "Intuitionistic logic, model theory and forcing" , North-Holland  (1969)</TD></TR></table>



====Comments====
For a modern elementary introduction with applications see [[#References|[a1]]]. A quicker, though incomplete, account is [[#References|[a3]]]. [[#References|[a2]]] presents the Boolean-valued approach mentioned above.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  K. Kunen,   "Set theory, an introduction to independence proofs" , North-Holland  (1980)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.L. Bell,   "Boolean-valued models and independence proofs in set theory" , Oxford Univ. Press  (1977)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J.P. Burgess,   "Forcing"  J. Barwise (ed.) , ''Handbook of mathematical logic'' , North-Holland  (1977)  pp. 403–452</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  T.J. Jech,   "Set theory" , Acad. Press  (1978)  pp. 523ff  (Translated from German)</TD></TR></table>