forcing

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  .

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 $\mathfrak{M}$ be a transitive     model of set theory, which we call the ground model. We want to “add a new set” $S$ to $\mathfrak{M}$ in such a way that the extension  $\mathfrak{M}^{\prime}$ has $\mathfrak{M}$ as a subclass, and the properties of $\mathfrak{M}$ are preserved, and $S\in\mathfrak{M}^{\prime}$.

The first step is to “approximate” the new set using elements of $\mathfrak{M}$. 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 $p\leq q$ 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 $\mathfrak{M}$.

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 $d\leq p$. A filter in $P$ is said to be $\mathfrak{M}$-generic if and only if it intersects every one of the dense subsets of $P$ which are in $\mathfrak{M}$. An $\mathfrak{M}$-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 $\mathfrak{M}$, generic filters are not elements of $\mathfrak{M}$.

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

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

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 $\mathfrak{M}$ be a transitive model of ZFC, and let $(P,\leq)$ be the set (in $\mathfrak{M}$) of all functions  $f$ whose domain is a finite subset of $\aleph_{2}\times\aleph_{0}$, and whose range is the set $\{0,1\}$. The ordering here is $p\leq q$ 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 $\aleph_{2}\times\aleph_{0}$, and range is $\{0,1\}$. We can see this $f$ as coding $\aleph_{2}$ new functions $f_{\alpha}:\aleph_{0}\to\{0,1\}$, $\alpha<\aleph_{2}$, which are subsets of omega. These functions are all distinct. $(P,\leq)$ http://planetmath.org/node/3242doesn’t collapse cardinals since it satisfies the countable chain condition. Thus $\aleph_{2}^{\mathfrak{M}[G]}=\aleph_{2}^{\mathfrak{M}}$ and CH is false in $\mathfrak{M}[G]$.

All this relies on a proper definition of the satisfaction relation in $\mathfrak{M}[G]$, 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