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 (poset) $(\mathbb{P},\leq)\in M$ (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 $M[G]$. 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 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.
In particular, an inner model (a class-sized transitive model (of ZFC or a weaker theory) containing all ordinals) can be a ground of $V$.
Contents
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 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 $y$". 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. The subsection below provides many more examples of consistency results, where the theory $T_1$ above is often $\text{ZFC}$ augmented by large cardinal axioms.
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: It is consistent relative to the existence of a $(\delta+2)-$strong cardinal $\delta$ that $2^\kappa=\kappa^+$ for every successor $\kappa$ but $2^\kappa=\kappa^{++}$ for every limit cardinal $\kappa$.
- Violating the Generalized Continuum Hypothesis everywhere: It is consistent relative to the existence of a $(\delta+2)-$strong cardinal $\delta$ that $2^\kappa=\kappa^{++}$ 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 (sub)model (of a generic extension) satisfying $\text{ZF(+}\neg\text{AC)}$ in which $\kappa=\aleph_1$ and $\omega_1$ is measurable/supercompact/huge (by the ultrafilter characterizations, not by the elementary embedding characterizations.)
- Singularity of every uncountable cardinal: It is consistent relative to the existence of a proper class of strongly compact cardinals there is model of $\text{ZF}$ in which (the axiom of choice does not hold and) every uncountable cardinal is singular and has cofinality $\omega$. The existence of a such model 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 $\kappa$. Then there is a (sub)model (of a generic extension) that satisfies $\text{ZF+DC+}\neg\text{AC}$ and in which $\kappa=2^{\aleph_0}$ and 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 $\omega_2$-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:
- It is consistent relative to the existence of an inaccessible cardinal that there are no Kurepa trees.
- Let $\kappa$ be a superstrong cardinal. Let $V[G]$ be the generic extension of $V$ by the Lévy collapse $\mathrm{Coll}(\aleph_0,<\kappa)$. 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 $\omega_2$-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 $\omega_2$ has the tree property. In fact, if there is infinitely many weakly compact cardinals then in a generic extension $\omega_{2n}$ has the tree property for every $n$. [1]
- It is consistent relative to the existence of infinitely many supercompact cardinals that 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 any such model.
- 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 $\kappa=2^{\aleph_0}$".
- It is consistent relative to a supercompact cardinal that there is an inaccessible cardinal $\kappa$, a cardinal $\lambda>\kappa$ and a stationary set $S\subseteq\mathcal{P}_\kappa(\lambda)$ that cannot be partitioned into $\kappa^+$ disjoint stationary subsets.
Types of forcing
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) (or just Knaster property) if every uncountable set of conditions has an uncountable subet of pairwise compatible elements. Every notion with property (K) satisfies the c.c.c.[1]
A poset is productive-ccc, if its product with any ccc poset is also ccc (in short $\textit{Prod-ccc}$).
A ccc poset $\mathbb{P}$ is strongly-$\underset{\sim}{Σ_n}$ if it is $Σ_n$-definable in $H(ω_1)$ with parameters, and the predicate “$x$ codes a maximal antichain of $\mathbb{P}$” is also $Σ_n$-definable in $H(ω_1)$ with parameters.[1]
A projective poset $\mathbb{P}$ is absolutely-ccc if it is ccc in every inner model $W$ of $V$ which satisfies ZFC and contains the parameters of the definition of $\mathbb{P}$.[1]
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.
Axiom A
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 $\omega_1$: if $\mathbb{P}$ is proper then every countable set of ordinals in $M[G]$ is a subset of a countable set in $M$.
Semi-proper forcing TODO
A projective poset $\mathbb{P}$ is strongly-proper if for every countable transitive model $N$ of a fragment of ZFC with the parameters of the definition of $\mathbb{P}$ in $N$ and such that $(\mathbb{P}^N, ≤_\mathbb{P}^N, ⊥_\mathbb{P}) ⊆ (\mathbb{P}, ≤_\mathbb{P}, ⊥_\mathbb{P})$, and for every $p ∈ \mathbb{P}^N$, there is $q ≤ p$ which is $(N, \mathbb{P})$-generic, i.e., if $N \models$ “$A$ is a maximal antichain of $\mathbb{P}$”, then $A ∩ N$ is predense below $q$.[1]
- A projective strongly-proper poset is proper.
The Lévy collapse
Prikry forcing
Other types of forcing, relations
(subsection from [1])
Among important classes of posets are:
- $σ$-centered posets (can be partitioned into countably many classes so that each class is finite-wise compatible)
- $σ$-linked posets (can be partitioned into countably many classes so that each class is pair-wise compatible)
- posets that preserve stationary subsets of $ω_1$ (in short $\textit{Stat-pres}$)
- posets that preserve $ω_1$ (in short $\textit{$ω_1$-pres}$)
We have
- $\textit{$\sigma$-centered} \subset \textit{$\sigma$-linked} \subset \textit{Knaster} \subset \textit{Prod-ccc} \subset \textit{ccc} \subset \textit{Proper} \subset \textit{Semi-proper} \subset \textit{Stat-pres} \subset \textit{$ω_1$-pres}$
Besides:
- the poset for adding a Cohen real ($\textit{Cohen}$, compare subsection above)
- the poset for adding a random real ($\textit{Random}$)
- the amoeba poset for measure ($\textit{Amoeba}$)
- the amoeba poset for category ($\textit{Amoeba-category}$)
- the Hechler forcing for adding a dominating real
- the Mathias forcing
- Borel forcing notions (the set of conditions is a Borel set)
- the $σ$-linked forcing notion for adding $ω_1$ random reals ($\textit{$ω_1$-Random}$)
Product forcing
Iterated forcing
Forcing axioms
Martin's axiom
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 its negation. $\text{MA}_{\aleph_1}$ 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.
Proper Forcing Axiom
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 that supercompact is $\aleph_2$ and $\text{PFA}$ 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 $\kappa$, therefore it implies the axiom of quasi-projective determinacy. 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 in combinatorial set theory.
Statement equivalent to $\text{PFA}$: If $\mathcal{M} = (M ; ∈, (R_i | i < ω_1 ))$ is a transitive model, $φ(x)$ is a $Σ_1$-formula and $\mathbb{Q}$ is a proper forcing such that $\Vdash_\mathbb{Q} φ(\mathcal{M})$, then there is in $V$ a transitive $\bar{\mathcal{M}} = (\bar{M} ; ∈, (\bar{R}_i | i < ω_1 ))$ together with an elementary embedding $j : \bar{\mathcal{M}} → \mathcal{M}$ such that $φ(\bar{\mathcal{M}})$ holds.[2]
Martin's maximum and the semiproper forcing axiom
Martin's Maximum is a strengthening of the proper forcing axiom defined the following way: suppose $(\mathbb{P},\leq)$ is a forcing notion that preserves stationary subsets of $\omega_1$, and that $\mathcal{D}$ is a collection of $\aleph_1$ dense subsets of $\mathbb{P}$, then there is a $\mathcal{D}$-generic filter on $\mathbb{P}$. This implies the proper forcing axiom, and also that the nonstationary ideal on $\omega_1$ is $\aleph_2$-saturated. It also implies that for all $\kappa\geq\aleph_2$, if $\kappa$ is regular then $\kappa^{\aleph_0}=\kappa$.
Bounded Forcing Axiom
(Section from [1])
The Bounded Forcing Axiom for a partial ordering $\mathbb{P}$, in short $BFA(\mathbb{P})$, is the statement:
- For every collection $\{I_α : α < ω_1 \}$ of maximal antichains of $\mathbf{B} \overset{def}{=} r.o.(\mathbb{P}) \setminus \{\mathbf{0}\}$, each of size $≤ ω_1$, there is a filter $G ⊆ \mathbf{B}$ such that for every $α$, $I_α ∩ G ≠ ∅$.
For a class of posets $Γ$, $BFA(Γ)$ is the statement that for every $\mathbb{P} ∈ Γ$, $BFA(\mathbb{P})$.
Examples:
- $MA_{ω_1}$, Martin’s axiom for $ω_1$, is $BFA(\textit{ccc})$.
- $BPFA$, the bounded proper forcing axiom, is $BFA(\textit{Proper})$.
- $BSPFA$, the bounded semi-proper forcing axiom, is $BFA(\textit{Semi-proper})$.
- $BMM$, the bounded Martin’s maximum, is $BFA(\textit{Stat-pres})$.
$BPFA$ implies that there is a well-ordering of the reals in length $ω_2$ definable with parameters in $H(ω_2)$ and therefore $\mathfrak{c} = \aleph_2$.
$BMM$ implies that for every set $X$ there is an inner model with a strong cardinal containing $X$.
- Thus, in particular, $BMM$ implies that for every set $X$, $X^\dagger$ exists.
$\text{wPFA}$ and $\text{PFA}_κ$
(information in this subsection from [2])
The weak Proper Forcing Axiom is obtained by requiring only that embedding $j$ (like in the last statement equivalent to $\text{PFA}$) exists in a forcing extension: If $\mathcal{M} = (M ; ∈, (R_i | i < ω_1 ))$ is a transitive model, $φ(x)$ is a $Σ_1$-formula and $\mathbb{Q}$ is a proper forcing such that $\Vdash_\mathbb{Q} φ(\mathcal{M})$, then there is a transitive $\bar{\mathcal{M}} = (\bar{M} ; ∈, (\bar{R}_i | i < ω_1 ))$ in $V$ and an elementary embedding $j : \bar{\mathcal{M}} → \mathcal{M}$ in a set-forcing extension (equivalently in $V^{Coll(ω, \bar{M})}$) such that $φ(\bar{\mathcal{M}})$ holds.
If there is a remarkable cardinal, then $\text{wPFA}$ holds in a forcing extension by a proper poset. If $\text{wPFA}$ holds, then $ω_2^V$ is remarkable in $L$.
For a cardinal $κ$, $\text{PFA}_κ$ is the statement that
- if $\mathbb{B}$ is any proper complete Boolean algebra and if $\langle A_ξ | ξ < ω_1 \rangle$ is any family of maximal antichains in $\mathbb{B}$ with $|A_ξ| ≤ κ$ for each $ξ < ω_1$, then there is some filter $G ⊆ \mathbb{B}$ such that $\forall_{ξ < ω_1} G ∩ A_ξ ≠ ∅$.
Equivalently, in analogy to the other statements (adding the assumption $|M| ≤ κ$):
- If $\mathcal{M} = (M ; ∈, (R_i | i < ω_1 ))$ is a transitive model, $|M| ≤ κ$, $φ(x)$ is a $Σ_1$-formula and $\mathbb{Q}$ is a proper forcing such that $\Vdash_\mathbb{Q} φ(\mathcal{M})$, then there is in $V$ a transitive $\bar{\mathcal{M}} = (\bar{M} ; ∈, (\bar{R}_i | i < ω_1 ))$ together with an elementary embedding $j : \bar{\mathcal{M}} → \mathcal{M}$ such that $φ(\bar{\mathcal{M}})$ holds.
$\text{PFA}_{\aleph_1}$ is $\text{BPFA}$.
$\text{wPFA}$ implies $\text{PFA}_{\aleph_2}$. However, it does not imply $\text{PFA}_{\aleph_3}$, because assertion $\text{wPFA} ∧ ∀_{κ ≥ \aleph_2} \square_κ$ is consistent relative to a remarkable cardinal and (Todorcevic, 1984, Theorem 1) $\text{PFA}_{\aleph_3}$ implies the failure of $\square_{ω_2}$.
Generic absoluteness and universal Baireness
Main article: Universally Baire
Generic ultrapowers
Main article: Precipitous ideals
Axioms of generic absoluteness
Main article: Axioms of generic absoluteness
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References
- Bagaria, Joan. Axioms of generic absoluteness. Logic Colloquium 2002 , 2006. www DOI bibtex
- Bagaria, Joan and Gitman, Victoria and Schindler, Ralf. Generic {V}opěnka's {P}rinciple, remarkable cardinals, and the weak {P}roper {F}orcing {A}xiom. Arch Math Logic 56(1-2):1--20, 2017. www DOI MR bibtex