# Forcing

Forcing is a method for extending a transitive model $M$ of $\text{ZFC}$ (the ground 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 $q$" or as "$p$ implies $q$". $G$ 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 $M[G]$.

Forcing was first introduced by Paul Cohen as a way of proving the consistency of the failure of the continuum hypothesis with $\text{ZFC}$. He also used it to prove the consistency of the failure of the axiom of choice, 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.

## 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.

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 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. $D\cap G\neq\empty$.

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:

• $V^\mathbb{P}_0=\empty$, $V^\mathbb{P}_\lambda = \bigcup_{\alpha<\lambda}V^\mathbb{P}_\alpha$ 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 there is a $M$-generic filter $G$ with $p\in G$ such that $M[G]\models\varphi(x_1^G,...,x_n^G)$. 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 any $M$-generic $G\ni p$ is such that $M[G]\models\varphi(x_1^G,...,x_n^G)$ (i.e. if $p\Vdash_{M,\mathbb{P}}\varphi(x_1,...,x_n)$) then all $M$-generic filters also produces a model satisfying $\varphi(x_1^G,...,x_n^G)$.

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 $y$". Also $i(x)=[x]$. It is sometimes convenient to identify a forcing notion with its separative quotient.

## 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 $\aleph_1$-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 $G\subseteq\mathbb{P}$.

$(\mathbb{P},\leq)$ 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.

$(\mathbb{P},\leq)$ 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.

## 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 $M$'s regular cardinals to the set of $M$'s cardinals, such that for all regular $\kappa$, $\mathrm{cf}F(\kappa)>\kappa$. Then there is a generic extension $M[G]$ of $M$ with the same cardinals and cofinalities such that $M[G]\models\text{ZFC+}\forall\kappa($if $\kappa$ is regular then $2^\kappa=F(\kappa)$).
• Violating the Singular Cardinal Hypothesis at $\aleph_\omega$: Assume there is a measurable cardinal of Mitchell order $o(\kappa)=\kappa^{++}$. Then there is a generic extension in which $\kappa=\aleph_\omega$ and $2^{\aleph_\omega}=\aleph_{\omega+2}$. The hypothesis used here is optimal: in term of consistency strength, no less than a measurable of order $\kappa^{++}$ can produce a model where $\text{SCH}$ fails.
• Violating the Singular Cardinal Hypothesis everywhere: Assume there is a strong cardinal. Then there is a generic extension in which $2^\kappa=\kappa^+$ for every successor $\kappa$ but $2^\kappa=\kappa^{++}$ for every limit cardinal $\kappa$.
• Violating the Generalized Continuum Hypothesis everywhere: Assume there exists a supercompact cardinal. Then there is a generic extension in which $2^\kappa=\kappa^{++}$ for every $\kappa$, i.e. $\text{GCH}$ fails everywhere.
• Large cardinal properties of $\aleph_1$: Let $\kappa$ be measurable/supercompact/huge. Then there is a generic extension satisfying $\text{ZF(+}\neg\text{AC)}$ in which $\kappa=\aleph_1$ and $\aleph_1$ is measurable/supercompact/huge (by the ultrafilter characterizations, not by the elementary embedding characterizations.)
• Singularity of every uncountable cardinal: Assume there is a proper class of strongly compact cardinals. Then there is a generic extension in which (the axiom of choice does not hold and) every uncountable cardinal is singular and has cofinality $\aleph_0$. This also implies that the axiom of determinacy holds in the $L(\mathbb{R})$ of some forcing extension of $\text{HOD}$.
• Regularity properties of all sets of reals: Assume there is an inaccessible cardinal. Then there is a generic extension satisfies $\text{ZF+DC+}\neg\text{AC}$ and in which every set of reals is Lebesgue measurable, has the Baire property and the perfect subset property. There is also a generic extension in which choice holds and every projective set of reals has those properties.
• Real-valued measurability of the continuum: Assume there is a measurable cardinal. Then there is a generic extension in which $\kappa=2^{\aleph_0}$ and $2^{\aleph_0}$ is real-valued measurable (and thus weakly inaccessible, weakly hyper-Mahlo, etc.)
• Precipitousness of the nonstationary ideal on $\omega_1$: Assume there is a measurable cardinal $\kappa$. Then there is a generic extension in which $\kappa=\aleph_1$ and the nonstationary ideal on $\omega_1$ is precipitous.
• Saturation of the nonstationary ideal on $\omega_1$: Assume there is a Woodin cardinal $\kappa$. Then there is a generic extension in which $\kappa=\aleph_2$ the nonstationary ideal on $\omega_1$ is $\aleph_2$-saturated.
• Saturation of an ideal on the continuum: Let $\kappa$ be a measurable cardinal. Then there is a generic extension in which $\kappa=2^{\aleph_0}$, there is a $2^{\aleph_0}$-saturated $2^{\aleph_0}$-complete ideal on $2^{\aleph_0}$ and there isn't any $\lambda$-saturated $2^{\aleph_0}$-complete ideal on $2^{\aleph_0}$ for every infinite $\lambda<2^{\aleph_0}$.

Some other applications of forcing:

• There is a generic extension in which there is a cardinal $\kappa$ such that $2^{\mathrm{cf}(\kappa)}<\kappa<\kappa^+<\kappa^{\mathrm{cf}(\kappa)}<2^\kappa$.
• Assume there is an inaccessible cardinal. Then there is a generic extension in which there are no Kurepa trees.
• Let $\kappa$ be a superstrong cardinal. Let $V[G]$ be the generic extension of $V$ by the Lévy collapse $\mathrm{Col}(\aleph_0,<\kappa)$. Then there is a nontrivial elementary embedding $j:L(\mathbb{R})\to(L(\mathbb{R}))^{V[G]}$.
• Let $\kappa$ be a superstrong cardinal. There exists a $\omega$-distributive $\kappa$-c.c. notion of forcing $(\mathbb{P},\leq)$ such that in $V^\mathbb{P}$, $\kappa=\aleph_2$ and there exists a normal $\aleph_2$-saturated $\sigma$-complete ideal on $\omega_1$.
• Let $\kappa$ be a weakly compact cardinal. Then there is a generic extension in which $\kappa=\aleph_2$ and $\aleph_2$ has the tree property. In fact, if there is infinitely many weakly compact cardinals then in a generic extension $\aleph_{2n}$ has the tree property for every $n$. [1]
• Assume there are infinitely many supercompact cardinals. Then there is a generic extension in which there exists infinitely many cardinals $\delta$ above $2^{\aleph_0}$ such that both $\delta$ and $\delta^+$ have the tree property. Also, the axiom of projective determinacy holds in that generic extension.
• Let $\kappa$ be a measurable cardinal. Then there is a generic extension in which $\kappa$ remains weakly compact, there is a $\kappa^+$-saturated $\kappa$-complete ideal on $\kappa$ but there isn't any $\kappa$-saturated $\kappa$-complete ideal on $\kappa$. One can replace "$\kappa$ is weakly compact" by "$\kappa$ is weakly inaccessible and $\kappa=2^{\aleph_0}$".

## Forcing axioms

Martin's axiom ($\text{MA}$) is the following assertion: If $(\mathbb{P},\leq)$ is a forcing notion that satisfies the countable chain condition and if $\mathcal{D}$ is a collection of fewer than $2^{\aleph_0}$ dense subsets of $\mathbb{P}$, then there exists a $\mathcal{D}$-generic filter on $\mathbb{P}$. By replacing "fewer than $2^{\aleph_0}$" by "at most $\kappa$" on obtain the axiom $\text{MA}_\kappa$. Martin's axiom is then $\text{MA}_{<2^{\aleph_0}}$. Note that $\text{MA}_{\aleph_0}$ is provably true in $\text{ZFC}$.

For all $\kappa$, $\text{MA}_\kappa$ implies $\kappa<2^{\aleph_0}$. Martin's axiom follows from the continuum hypothesis, but is also consistent with the continuum hypothesis. $\text{MA}_{\aleph_1}$ implies there are no Suslin trees, that every Aronszajn tree is special, and that the c.c.c. is equivalent to property (K).

Martin's axiom implies that $2^{\aleph_0}$ is regular, that it is not real-valued measurable, and also that $2^\lambda=2^{\aleph_0}$ for all $\lambda<2^{\aleph_0}$. It implies that the intersection of fewer than $2^{\aleph_0}$ dense open sets is dense, the union of fewer than $2^{\aleph_0}$ null sets is null, and the union of fewer than $2^{\aleph_0}$ meager sets is meager. Also, the Lebesgue measure is $2^{\aleph_0}$-additive. If additionally $\neg\text{CH}$ then every $\mathbf{\Sigma}^1_2$ set is Lebesgue measurable and has the Baire property.

A forcing notion $(\mathbb{P},\leq)$ satisfies Axiom A if there is a sequence of partial orderings $\{\leq_n:n<\omega\}$ of $\mathbb{P}$ such that $p\leq_0 q$ implies $p\leq q$, for all n $p\leq_{n+1} q$ implies $p\leq_n q$, and the following conditions holds:

• for every descending sequence $p_0\geq_0 p_1\geq_1...\geq_{n-1}p_n\geq_n...$ there is a $q$ such that $q\leq_n p_n$ for all $n$.
• for every $p\in\mathbb{P}$, for every $n$ and every ordinal name $\alpha$ there exists $q\leq_n p$ and a countable set $B$ such that $p\Vdash\alpha\in B$.

Every c.c.c. or $\omega$-closed notion satisfies Axiom A.

### Proper forcing

We say that a forcing notion $(\mathbb{P},\leq)$ is proper if for every uncountable cardinal $\lambda$, every stationary subset of $[\lambda]^\omega$ remains stationary in every generic extension. Every c.c.c. or $\omega$-closed notion is proper, and so is every notion satisfying Axiom A. Proper forcing does not collapse $\aleph_1$: if $\mathbb{P}$ is proper then every countable set of ordinals in $M[G]$ is a subset of a countable set in $M$.

The Proper Forcing Axiom ($\text{PFA}$) is obtained by replacing "c.c.c." by "proper" in the definition of $\text{MA}_{\aleph_1}$: for every proper forcing notion $(\mathbb{P},\leq)$, if $\mathcal{D}$ is a collection of $\aleph_1$ dense subsets of $\mathbb{P}$, then there is a $\mathcal{D}$-generic filter on $\mathbb{P}$. $\text{PFA}$ implies $2^{\aleph_0}=\aleph_2$ and that the continuum (i.e. $\aleph_2$) has the tree property. It also implies that every two $\aleph_1$-dense sets of reals are isomorphic.

Unlike Martin's axiom, which is equiconsistent with $\text{ZFC}$, the $\text{PFA}$ has very high consistency strength, slightly below that of a supercompact cardinal. If there is a supercompact cardinal then there is a generic extension in which $\text{PFA}$ holds. On the other hand, [2] proves a quasi lower bound on the consistency strength of the $\text{PFA}$, which is at least the existence of a proper class of subcompact cardinals. [3] also shows that all known methods of forcing $\text{PFA}$ requires a strongly compact cardinal, and if one wants the forcing to be proper, a supercompact is required.

$\text{PFA}$ implies the failure of the square principle $\Box_\kappa$ for every uncountable cardinal $\kappa$. It also implies the Open Coloring Axiom: let $X$ be a set of reals, and let $K\subseteq[X]^2$. We say that $K$ is open if the set $\{(x,y):\{x,y\}\in K\}$ is an open set in the space $X\times X$. Then

• ($\text{OCA}$). For every $X\subseteq\mathbb{R}$, and for any partition $[X]^2=K_0\cup K_1$ with $K_0$ open, either there exists an uncountable $Y\subseteq X$ such that $[Y]^2\subseteq K_0$ or there exists sets $H_n, n<\omega$ such that $X=\bigcup_{n<\omega}H_n$ and $[H_n]^2\subseteq K_1$ for all $n$.

This axiom has many useful implications combinatorial set theory.

## Other types of forcing

    This article is a stub. Please help us to improve Cantor's Attic by adding information.