**Forcing** is a method for extending a transitive model $M$ of
$\text{ZFC}$ (the *ground
[model](Model.md "Model")*) by
adjoining a new set $G$ (the *generic set*) to produce a new, larger
model $M[G]$ called a *generic extension*. In short, the set $G$ can
be constructed a certain way using a partially ordered set
$(\mathbb{P},\leq)\in M$ (the *forcing notion*) so that the following
holds:
- **(Generic Model Theorem).** There exists a unique transitive model
$M[G]$ of $\text{ZFC}$ that includes $M$ (as a subset) and
contains $G$ (as an element), has the same ordinals as $M$, and any
transitive model of $\text{ZFC}$ also including $M$ and containing
$G$ includes $M[G]$ (i.e. $M[G]$ is minimal).
The elements of the forcing notion $\mathbb{P}$ will be called the
*conditions*. The order $p\leq q$, for $p,q\in\mathbb{P}$, is to be
interpreted as "$p$ is stronger than $q
quot; or as "$p$ implies $qquot;. $G$
will be a special subset of $\mathbb{P}$ said to be *generic over $M$*
and satisfying some requirements. The choice of $\mathbb{P}$ and of
$\leq$ will determine what is true of false in $M[G]$. A special
relation called the *forcing relation* is defined, which links the
conditions to the formulas they will force. It is very important to note
that this relation can be defined from within the ground model $M$.
While the usual definition of forcing only works for transitive
countable models $M$ of $\text{ZFC}$, it is customary to "take $V$ as
the ground model", and pretend there exists a generic
$G\subseteq\mathbb{P}$. Every statement about the generic extension
$V[G]$ can be thought as a statement in the forcing relation: that
relation being definable within the ground model, this method can be
thought as working within the ground model $M$, with $V[G]$ being, in
some way, $M[G]$ as seen from within the ground model $M$.
Forcing was first introduced by Paul Cohen as a way of proving the
consistency of the failure of the
<a href="GCH" class="mw-redirect" title="GCH">continuum hypothesis</a>
with $\text{ZFC}$. He also used it to prove the consistency of the
failure of the
<a href="Axiom_of_choice" class="mw-redirect" title="Axiom of choice">axiom of choice</a>,
albeit the proof is more indirect: if $M$ satisfies choice, then so does
$M[G]$, so $\neg AC$ cannot be forced directly, though it is possible
to extract a submodel of $M[G]$ (for a particular generic extension)
in which choice fails.
In particular, an inner model can be a ground of $V$.
## Definitions and some properties
Let $(\mathbb{P},\leq)$ be a partially ordered set, the *forcing
notion*. Sometimes $\leq$ can just be a preorder (i.e. not necessarily
antisymmetric). The elements of $\mathbb{P}$ are called *conditions*.
We will assume $\mathbb{P}$ has a maximal element $1$, i.e. one has
$p\leq 1$ for all $p\in\mathbb{P}$. This element isn't necessary if
one uses the definition using Boolean algebras presented below, but is
useful when trying to construct $M[G]$ without using Boolean algebras. ^e01f21
Two conditions $p,q\in P$ are *compatible* if there exists
$r\in\mathbb{P}$ such that $r\leq p$ and $r\leq q$. They are
*incompatible* otherwise. A set $W\subseteq\mathbb{P}$ is an
*antichain* if all its elements are pairwise incompatible.
### Genericity
A nonempty set $F\subseteq\mathbb{P}$ is a
*[filter](Filter.md "Filter")
on $\mathbb{P}$* if all of its elements are pairwise compatible and it
is closed under implications, i.e. if $p\leq q$ and $p\in F$ then
$q\in F$.
One says that a set $D\subseteq\mathbb{P}$ is *dense* if for all
$p\in\mathbb{P}$, there is $q\in D$ such that $q\leq p$ (i.e. $q$
*implies* $p$). $D$ is *open dense* if additionally $q\leq p$ and
$p\in D$ implies $q\in D$. $D$ is *predense* if every
$p\in\mathbb{P}$ is compatible with some $q\in D$.
Given a collection $\mathcal{D}$ of dense subsets of $\mathbb{P}$, one
says that a filter $G$ is **$\mathcal{D}$-generic** if it intersects
all sets $D\in\mathcal{D}$, i.e. $D\cap G\neq\varnothing$.
Given a transitive model $M$ of $\text{ZFC}$ such that
$(\mathbb{P},<)\in M$, we say that a filter
$G\subseteq\mathbb{P}$ is **$M$-generic** (or $\mathbb{P}$-generic in
$M$, or just generic) if it is $\mathcal{D}_M$-generic where
$\mathcal{D}_M$ is the set of all dense subsets of $\mathbb{P}$ in
$M$.
In the above definitions, *dense* can be replaced with *open dense*,
*predence* or *a maximal antichain*, and the resulting notion of
genericity would be the same.
In most cases, if $G$ is $\mathbb{P}$-generic over $M$ then $G\not\in
M$. The Generic Model Theorem mentioned above says that there is a
minimal model $M[G]\supseteq M$ with $M[G]\models\text{ZFC}$,
$G\in M[G]$, and if $M\models$ "$x$ is an ordinal" then so does
$M[G]$.
### $\mathbb{P}$-names and interpretation by $G$
Using transfinite recursion, define the following cumulative hierarchy:
- $V^\mathbb{P}_0=\varnothing$, $V^\mathbb{P}_\lambda =
\bigcup_{\alpha<\lambda}V^\mathbb{P}_\alpha$ for limit
$\lambda$
- $V^\mathbb{P}_{\alpha+1} =
\mathcal{P}(V^\mathbb{P}_\alpha\times\mathbb{P})$
- $V^\mathbb{P} =
\bigcup_{\alpha\in\mathrm{Ord}}V^\mathbb{P}_\alpha$
Elements of $V^\mathbb{P}$ are called *$\mathbb{P}$-names*. Every
nonempty $\mathbb{P}$-name is of a set of pairs $(n,p)$ where $n$ is
another $\mathbb{P}$-name and $p\in\mathbb{P}$.
Given a filter $G\subseteq\mathbb{P}$, define the *interpretation of
$\mathbb{P}$-names* by $G$: Given a $\mathbb{P}$-name $x$, let
$x^G=\{y^G : ((\exists p\in G)(y,p)\in x)\}$. Letting
$\breve{x}=\{(\breve{y},1):y\in x\}$ for every set $x$ be the
*canonical name* for $x$, one has $\breve{x}^G=x$ for all $x$.
Let $M$ be a transitive model of $\text{ZFC}$ such that
$(\mathbb{P},\leq)\subseteq M$. Let $M^\mathbb{P}$ be the
$V^\mathbb{P}$ constructed in $M$. Given a $M$-generic filter
$G\subseteq\mathbb{P}$, we can now define the generic extension
$M[G]$ to be $\{x^G : x\in M^\mathbb{P}\}$. This $M[G]$
satisfies the Generic Model Theorem.
### The forcing relation
Define the *forcing language* to be the first-order language of set
theory augmented by a constant symbol for every $\mathbb{P}$-name in
$M^\mathbb{P}$. Given a condition $p\in\mathbb{P}$, a formula
$\varphi(x_1,...,x_n)$ and $x_1,...,x_n \in M^\mathbb{P}$, we say
that **$p$ forces $\varphi(x_1,...,x_n)$**, denoted $p\Vdash_
{M,\mathbb{P}}\varphi(x_1,...,x_n)$ if for all $M$-generic filter
$G$ with $p\in G$ one has $M[G]\models\varphi(x_1^G,...,x_n^G)$.
There exists an "internal" definition of $\Vdash$, i.e. a definition
formalizable in $M$ itself, by induction on the complexity of the
formulas of the forcing language.
The **Forcing Theorem** asserts that if $\sigma$ is a sentence of the
forcing language, $M[G]$ satisfies $\sigma$ if and only if some
condition $p\in G$ forces $\sigma$.
The forcing relation has the following properties, for all
$p,q\in\mathbb{P}$ and formulas $\varphi,\psi$ of the forcing
language:
- $p\Vdash\varphi\land q\leq p\implies q\Vdash\varphi$
- $p\Vdash\varphi\implies\neg(p\Vdash\neg\varphi)$
- $p\Vdash\neg\varphi\iff\neg\exists q\leq p(q\Vdash\varphi)$
- $p\Vdash(\varphi\land\psi)\iff(p\Vdash\varphi\land
p\Vdash\psi)$
- $p\Vdash\forall x\varphi\iff\forall x\in
M^\mathbb{P}(p\Vdash\varphi(x))$
- $p\Vdash(\varphi\lor\psi)\iff\forall q\leq p\exists r\leq
q(r\Vdash\varphi\lor r\Vdash\psi)$
- $p\Vdash\exists x\varphi\iff\forall q\leq p\exists r\leq
q\exists x\in M^\mathbb{R}(r\Vdash\varphi(x))$
- $p\Vdash\exists x\varphi\implies\exists x\in
M^\mathbb{P}(p\Vdash\varphi(x))$
- $\forall p\exists q\leq p (q\Vdash\varphi\lor
q\Vdash\neg\varphi)$
### Separativity
A forcing notion $(\mathbb{P},\leq)$ is *separative* if for all
$p,q\in\mathbb{P}$, if $p\not\leq q$ then there exists a $r\leq p$
incompatible with $q$. Many notions aren't separative, for example if
$\leq$ is a linear order than $(\mathbb{P},\leq)$ is separative iff
$\mathbb{P}$ has only one element. However, every notion
$(\mathbb{P},\leq)$ has a unique (up to isomorphism) *separative
quotient* $(\mathbb{Q},\preceq)$, i.e. a notion
$(\mathbb{Q},\preceq)$ and a function $i:\mathbb{P}\to\mathbb{Q}$
such that $x\leq y\implies i(x)\preceq i(y)$ and $x, y$ are
compatible iff $i(x),i(y)$ are compatible. This name comes from the fact
that $\mathbb{Q}=(\mathbb{P}/\equiv)$ where $x\equiv y$ iff every
$z\in P$ is compatible with $x$ iff it is compatible with $y$. The
order $\preceq$ on the equivalence classes is "$[x]\preceq[y]$ iff
all $z\leq x$ are compatible with $yquot;. Also $i(x)=[x]$. It is
sometimes convenient to identify a forcing notion with its separative
quotient.
## Boolean algebras
*To be expanded.*
It is sometimes convenient to take the forcing notion $\mathbb{P}$ to
be a Boolean algebra $\mathbb{B}$.
## Consistency proofs
Let $T_1$ and $T_2$ be some recursively enumerable enumerable
extensions of $\text{ZFC}$ (possibly $\text{ZFC}$ itself). The
existence of a countable transitive model $M$ of the theory $T_1$ is
equivalent to the assertion that $T_1$ is consistent. When we construct
a generic extension $M[G]$ satisfying $T_2$ from a countable
transitive model $M$ of $T_1$, we prove the consistency of $T_2$
(since we prove it has a set model) only from the consistency of $T_1$,
i.e. we prove $\text{Con}(T_1)\implies\text{Con}(T_2)$.
For instance, by following Cohen's construction of a generic extension
statisfying $\text{ZFC+}\neg\text{CH}$ from a model of $\text{ZFC}$,
we prove that
$\text{Con}(\text{ZFC})\implies\text{Con}(\text{ZFC+}\neg\text{CH})$.
It follows that if $\text{ZFC}$ is consistent then it cannot prove
$\text{CH}$, as otherwise $\text{ZFC+}\neg\text{CH}$ would be
inconsistent, contradicting the above implication proved by forcing.
Those implications between the consistencies of different theories are
the *relative consistency results* set theorists are often interested
in. A section way below provides many more examples of consistency
results, where the theory $T_1$ above is often $\text{ZFC}$ augmented
by large cardinal axioms.
## Chain conditions, distributivity, closure and property (K)
A forcing notion $(\mathbb{P},\leq)$ satisfies the *$\kappa$-chain
condition* ($\kappa$-c.c.) if every antichain of elements of
$\mathbb{P}$ has cardinality less than $\kappa$. The $\omega_1$-c.c.
is called the *countable chain condition* (c.c.c.). An important feature
of chain conditions is that if $(\mathbb{P},\leq)$ satisfies the
$\kappa$-c.c. then if $\kappa$ is regular in $M$ then it will be
regular in $M[G]$. Since the $\kappa$-c.c. implies the
$\lambda$-c.c. for all $\lambda\geq\kappa$, it follows that the
$\kappa$-c.c. implies all regular cardinals $\geq\|\mathbb{P}\|^+$
will be preserved, and in particular the c.c.c. implies all cardinals
and cofinalities of $M$ will be preserved in $M[G]$ for all
$M$-generic $G\subseteq\mathbb{P}$.
Let $\kappa$ be a regular uncountable cardinal. If
$(\mathbb{P},\leq)$ is a $\kappa$-c.c. notion of forcing then every
club $C\in M[G]$ of $\kappa$ has a subset $D$ that is a club subset
of $\kappa$ in the ground model; therefore every stationary subset of
$\kappa$ remains stationary in $M[G]$.
$(\mathbb{P},\leq)$ is *$\kappa$-distributive* if the intersection of
$\kappa$ open dense sets is still open dense. $\kappa$-distributive
notions for infinite $\kappa$ does not add new subsets to $\kappa$. A
stronger property, closure, is defined the following way: $\mathbb{P}$
is *$\kappa$-closed* if every $\lambda\leq\kappa$, every descending
sequence $p_0\geq p_1\geq...\geq p_\alpha\geq...
(\alpha<\lambda)$ has a lower bound. Every $\kappa$-closed notion
is $\kappa$-distributive. If, for some regular uncountable cardinal
$\kappa$ and all $\lambda<\kappa$, $(\mathbb{P},\leq)$ is a
$\lambda$-closed forcing notion, then every stationary subset of
$\kappa$ remains stationary in every generic extension.
$(\mathbb{P},\leq)$ has *property (K)* (K for Knaster) if every
uncountable set of conditions has an uncountable subet of pairwise
compatible elements. Every notion with property (K) satisfy the c.c.c.
## Cohen forcing, adding subsets of regular cardinals, and independence of the continuum hypothesis
Let $\kappa$ be a regular cardinal satisfying
$2^{<\kappa}=\kappa$. Let $\lambda>\kappa$ be a cardinal such
that $\lambda^\kappa=\lambda$. Let $\text{Add}(\kappa,\lambda) =
(\mathbb{P},\leq)$ be the following partial order: $\mathbb{P}$ is
the set of all functions $p$ with
$\text{dom}(p)\subseteq\lambda\times\kappa$,
$\|\text{dom}(p)\|<\kappa$ and $\text{ran}(p)\subseteq\{0,1\}$,
and let $p\leq q$ iff $p\supseteq q$. Let $G$ be a $V$-generic on
$\mathbb{P}$ and let $f=\bigcup G$. Then in $V[G]$, $f$ is a
function from $\lambda\times\kappa$ to $\{0,1\}$. For every
particular $\alpha<\lambda$, the function
$c_\alpha(\xi)=f(\alpha,\xi)$ is in $V[G]$ the characteristic
function of a subset
$x_\alpha=\{\xi<\kappa:c_\alpha(\xi)=1\}$ of $\kappa$. None
of those new subsets were originally in $V$, and if $\alpha\neq\beta$
then $x_\alpha\neq x_\beta$. Then, because $\mathbb{P}$ satisfies
the $\kappa^+$-chain condition, it follows that all cardinals are
preserved except that $2^\kappa=\lambda$.
In the special case $\kappa=\aleph_0$, there are new real numbers in
$V[G]$ and $2^{\aleph_0}=\lambda$. Those new real numbers are
called *Cohen reals*. This technique allows one to show that
$\text{ZFC}$ is consistent with the negation of the continuum
hypothesis, i.e. that $2^{\aleph_0}>\aleph_1$. In fact,
$2^{\aleph_0}$ can be any cardinal with uncountable cofinality, even
if singular, e.g. one can force $2^{\aleph_0}=\aleph_{\omega_1}$.
Note that $2^{\aleph_0}$ cannot be a cardinal of countable cofinality,
so this is impossible to force.
## Other examples of consistency results proved using forcing
In the following examples, the generated generic extensions satisfy the
axiom of choice unless indicated otherwise.
- **Easton's theorem:** Let $M$ be a transitive set model of
$\text{ZFC+GCH}$. Let $F$ be an increasing function in $M$ from the
set of $M