# forcing

Forcing^{} is the method used by Paul Cohen to prove the independence of
the continuum hypothesis^{} (CH). In fact, the method was used by Cohen to
prove that CH could be violated.

Adding a set to a model of set theory^{} via forcing is similar to adjoining a new element to a field. Suppose we have a field $k$, and we want to add to this field an
element $\alpha $ such ${\alpha}^{2}=-1$. We see
that we cannot simply drop a new $\alpha $ in $k$, since then we are
not guaranteed that we still have a field. Neither can we simply
assume that $k$ already has such an element. The standard way of
doing this is to start by adjoining a generic^{} indeterminate $X$, and
impose a constraint on $X$, saying that ${X}^{2}+1=0$. What we do is take
the quotient $k[X]/({X}^{2}+1)$, and make a field out of it by taking the
quotient field. We then obtain $k(\alpha )$, where $\alpha $ is the
equivalence class^{} of $X$ in the quotient.
The general case of this is the theorem of algebra saying that every
polynomial^{} $p$ over a field $k$ has a root in some extension field^{}.

We can rephrase this and say that “it is consistent with standard
field theory that $-1$ have a square root^{}”.

When the theory we consider is ZFC, we run in exactly the same
problem : we can’t just add a “new” set and pretend it has the
required properties, because then we may violate something else, like
foundation. Let $\U0001d510$ be a transitive^{} model of set theory, which we
call the ground model. We want to “add a new set” $S$ to $\U0001d510$ in
such a way that the extension^{} ${\U0001d510}^{\prime}$ has $\U0001d510$ as a subclass, and the
properties of $\U0001d510$ are preserved, and $S\in {\U0001d510}^{\prime}$.

The first step is to “approximate” the new set using elements of $\U0001d510$. This is the analogue of finding the irreducible polynomial in the algebraic example. The set $P$ of such “approximations” can be ordered by how much information the approximations give : let $p,q\in P$, then $p\le q$ if and only if $p$ “is stronger than” $q$. We call this set a set of forcing conditions. Furthermore, it is required that the set $P$ itself and the order relation be elements of $\U0001d510$.

Since $P$ is a partial order^{}, some of its subsets have interesting
properties. Consider $P$ as a topological space^{} with the order
topology. A subset $D\subseteq P$ is dense in $P$ if and only
if for every $p\in P$, there is $d\in D$ such that $d\le p$. A
filter in $P$ is said to be $\U0001d510$-generic if and only if it intersects
every one of the dense subsets of $P$ which are in $\U0001d510$. An $\U0001d510$-generic
filter in $P$ is also referred to as a generic set of conditions
in the literature. In general, even though $P$ is a set in $\U0001d510$, generic filters are not elements of $\U0001d510$.

If $P$ is a set of forcing conditions, and $G$ is a generic set of conditions in $P$, all in the ground model $\U0001d510$, then we define $\U0001d510[G]$ to be the least model of ZFC that contains $G$. The big theorem is this :

Theorem.
$\U0001d510[G]$ is a model of ZFC, and has the same ordinals^{} as $\U0001d510$, and
$\U0001d510\subseteq \U0001d510[G]$.

The way to prove that we can violate CH using a generic extension is
to add many new “subsets of $\omega $” in the following way : let
$\U0001d510$ be a transitive model of ZFC, and let $(P,\le )$ be the set (in
$\U0001d510$) of all functions^{} $f$ whose domain is a finite subset of
${\mathrm{\aleph}}_{2}\times {\mathrm{\aleph}}_{0}$, and whose range is the set $\{0,1\}$. The
ordering here is $p\le q$ if and only if $p\supset q$. Let
$G$ be a generic set of conditions in $P$. Then $\bigcup G$ is a
total function^{} whose domain is ${\mathrm{\aleph}}_{2}\times {\mathrm{\aleph}}_{0}$, and range is
$\{0,1\}$. We can see this $f$ as coding ${\mathrm{\aleph}}_{2}$ new functions
${f}_{\alpha}:{\mathrm{\aleph}}_{0}\to \{0,1\}$, $$,
which are subsets of omega. These
functions are all distinct. $(P,\le )$ http://planetmath.org/node/3242doesn’t collapse cardinals since it satisfies the countable chain condition. Thus ${\mathrm{\aleph}}_{2}^{\U0001d510[G]}={\mathrm{\aleph}}_{2}^{\U0001d510}$ and CH is false in $\U0001d510[G]$.

All this relies on a proper definition of the satisfaction relation in $\U0001d510[G]$, and the forcing relation. Details can be found in Thomas Jech’s book Set Theory.

Title | forcing |

Canonical name | Forcing |

Date of creation | 2013-03-22 12:44:17 |

Last modified on | 2013-03-22 12:44:17 |

Owner | ratboy (4018) |

Last modified by | ratboy (4018) |

Numerical id | 12 |

Author | ratboy (4018) |

Entry type | Definition |

Classification | msc 03E50 |

Classification | msc 03E35 |

Classification | msc 03E40 |

Related topic | ForcingRelation |

Related topic | CompositionOfForcingNotions |

Related topic | EquivalenceOfForcingNotions |

Related topic | FieldAdjunction |

Defines | forcing |