List of forcing notions

In mathematics, forcing is a method of constructing new models M[G] of set theory by adding a generic subset G of a poset P to a model M. The poset P used will determine what statements hold in the new universe (the 'extension'); to force a statement of interest thus requires construction of a suitable P. This article lists some of the posets P that have been used in this construction.

Notation

• P is a poset with order <
• V is the universe of all sets
• M is a countable transitive model of set theory
• G is a generic subset of P over M.

Definitions

• P satisfies the countable chain condition if every antichain in P is at most countable. This implies that V and V[G] have the same cardinals (and the same cofinalities).
• A subset D of P is called dense if for every p ∈ P there is some q ∈ D with q ≤ p.
• A filter on P is a nonempty subset F of P such that if p < q and p ∈ F then q ∈ F, and if p ∈ F and q ∈ F then there is some r ∈ F with r ≤ p and r ≤ q.
• A subset G of P is called generic over M if it is a filter that meets every dense subset of P in M.

Amoeba forcing

Amoeba forcing is forcing with the amoeba order, and adds a measure 1 set of random reals.

Cohen forcing

Main article: Cohen forcing

In Cohen forcing (named after Paul Cohen) P is the set of functions from a finite subset of ω₂ × ω to {0,1} and p < q if p ⊇ q.

This poset satisfies the countable chain condition. Forcing with this poset adds ω₂ distinct reals to the model; this was the poset used by Cohen in his original proof of the independence of the continuum hypothesis.

More generally, one can replace ω₂ by any cardinal κ so construct a model where the continuum has size at least κ. Here, there is no restriction. If κ has cofinality ω, the cardinality of the reals ends up bigger than κ.

Grigorieff forcing

Grigorieff forcing (after Serge Grigorieff) destroys a free ultrafilter on ω.

Hechler forcing

Hechler forcing (after Stephen Herman Hechler) is used to show that Martin's axiom implies that every family of less than c functions from ω to ω is eventually dominated by some such function.

P is the set of pairs (s, E) where s is a finite sequence of natural numbers (considered as functions from a finite ordinal to ω) and E is a finite subset of some fixed set G of functions from ω to ω. The element (s, E) is stronger than (t, F) if t is contained in s, F is contained in E, and if k is in the domain of s but not of t then s(k) > h(k) for all h in F.

Jockusch–Soare forcing

Forcing with $\Pi _{1}^{0}$ classes was invented by Robert Soare and Carl Jockusch to prove, among other results, the low basis theorem. Here P is the set of nonempty $\Pi _{1}^{0}$ subsets of $2^{\omega }$ (meaning the sets of paths through infinite, computable subtrees of $2^{<\omega }$), ordered by inclusion.

Iterated forcing

Main article: Iterated forcing

Iterated forcing with finite supports was introduced by Solovay and Tennenbaum to show the consistency of Suslin's hypothesis. Easton introduced another type of iterated forcing to determine the possible values of the continuum function at regular cardinals. Iterated forcing with countable support was investigated by Laver in his proof of the consistency of Borel's conjecture, Baumgartner, who introduced Axiom A forcing, and Shelah, who introduced proper forcing. Revised countable support iteration was introduced by Shelah to handle semi-proper forcings, such as Prikry forcing, and generalizations, notably including Namba forcing.

Laver forcing

Laver forcing was used by Laver to show that Borel's conjecture, which says that all strong measure zero sets are countable, is consistent with ZFC. (Borel's conjecture is not consistent with the continuum hypothesis.)

• P is the set of Laver trees, ordered by inclusion.

A Laver tree p is a subset of the finite sequences of natural numbers such that

• p is a tree: p contains any initial sequence of any element of p, equivalently stated as p is closed under initial segments
• p has a stem: a maximal node s(p) = s ∈ p such that s ≤ t or t ≤ s for all t in p,
• If t ∈ p and s ≤ t then t has an infinite number of immediate successors tn in p for n ∈ ω.

If G is generic for (P, ≤), then the real {s(p) : p ∈ G}, called a Laver-real, uniquely determines G.

Laver forcing satisfies the Laver property.

Levy collapsing

Main article: Collapsing algebra

These posets will collapse various cardinals, in other words force them to be equal in size to smaller cardinals.

• Collapsing a cardinal to ω: P is the set of all finite sequences of ordinals less than a given cardinal λ. If λ is uncountable then forcing with this poset collapses λ to ω.
• Collapsing a cardinal to another: P is the set of all functions from a subset of κ of cardinality less than κ to λ (for fixed cardinals κ and λ). Forcing with this poset collapses λ down to κ.
• Levy collapsing: If κ is regular and λ is inaccessible, then P is the set of functions p on subsets of λ × κ with domain of size less than κ and p(α, ξ) < α for every (α, ξ) in the domain of p. This poset collapses all cardinals less than λ onto κ, but keeps λ as the successor to κ.

Levy collapsing is named for Azriel Levy.

Magidor forcing

Amongst many forcing notions developed by Magidor, one of the best known is a generalization of Prikry forcing used to change the cofinality of a cardinal to a given smaller regular cardinal.

Mathias forcing

• An element of P is a pair consisting of a finite set s of natural numbers and an infinite set A of natural numbers such that every element of s is less than every element of A. The order is defined by

(t, B) is stronger than (s, A) ((t, B) < (s, A)) if s is an initial segment of t, B is a subset of A, and t is contained in s ∪ A.

Mathias forcing is named for Adrian Mathias.

Namba forcing

Namba forcing (after Kanji Namba) is used to change the cofinality of ω₂ to ω without collapsing ω₁.

• P is the set of all trees $T\subseteq \omega _{2}^{<\omega }$ (nonempty downward closed subsets of the set of finite sequences of ordinals less than ω₂) which have the property that any s in T has an extension in T which has $\aleph _{2}$ immediate successors. P is ordered by inclusion (i.e., subtrees are stronger conditions). The intersection of all trees in the generic filter defines a countable sequence which is cofinal in ω₂.

Namba' forcing is the subset of P such that there is a node below which the ordering is linear and above which each node has $\aleph _{2}$ immediate successors.

Magidor and Shelah proved that if CH holds then a generic object of Namba forcing does not exist in the generic extension by Namba', and vice versa.

Prikry forcing

In Prikry forcing (after Karel Prikrý) P is the set of pairs (s, A) where s is a finite subset of a fixed measurable cardinal κ, and A is an element of a fixed normal measure D on κ. A condition (s, A) is stronger than (t, B) if t is an initial segment of s, A is contained in B, and s is contained in t ∪ B. This forcing notion can be used to change to cofinality of κ while preserving all cardinals.

Product forcing

Taking a product of forcing conditions is a way of simultaneously forcing all the conditions.

• Finite products: If P and Q are posets, the product poset P × Q has the partial order defined by (p₁, q₁) ≤ (p₂, q₂) if p₁ ≤ p₂ and q₁ ≤ q₂.
• Infinite products: The product of a set of posets P_i, i ∈ I, each with a largest element 1 is the set of functions p on I with p(i) ∈ P(i) and such that p(i) = 1 for all but a finite number of i. The order is given by p ≤ q if p(i) ≤ q(i) for all i.
• The Easton product (after William Bigelow Easton) of a set of posets P_i, i ∈ I, where I is a set of cardinals is the set of functions p on I with p(i) ∈ P(i) and such that for every regular cardinal γ the number of elements α of γ with p(α) ≠ 1 is less than γ.

Radin forcing

Radin forcing (after Lon Berk Radin), a technically involved generalization of Magidor forcing, adds a closed, unbounded subset to some regular cardinal λ.

If λ is a sufficiently large cardinal, then the forcing keeps λ regular, measurable, supercompact, etc.

Random forcing

Main article: random algebra

• P is the set of Borel subsets of [0,1] of positive measure, where p is called stronger than q if it is contained in q. The generic set G then encodes a "random real": the unique real x_G in all rational intervals [r, s]^V[G] such that [r, s]^V is in G. This real is "random" in the sense that if X is any subset of [0, 1]^V of measure 1, lying in V, then x_G ∈ X.

Sacks forcing

• P is the set of all perfect trees contained in the set of finite {0, 1} sequences. (A tree T is a set of finite sequences containing all initial segments of its members, and is called perfect if for any element t of T there is a segment s extending t so that both s0 and s1 are in T.) A tree p is stronger than q if p is contained in q. Forcing with perfect trees was used by Gerald Enoch Sacks to produce a real a with minimal degree of constructibility.

Sacks forcing has the Sacks property.

Shooting a fast club

For S a stationary subset of $\omega _{1}$ we set $P=\{\langle \sigma ,C\rangle \,\colon \sigma$ is a closed sequence from S and C is a closed unbounded subset of $\omega _{1}\}$, ordered by $\langle \sigma ',C'\rangle \leq \langle \sigma ,C\rangle$ iff $\sigma '$ end-extends $\sigma$ and $C'\subseteq C$ and $\sigma '\subseteq \sigma \cup C$. In $V[G]$, we have that $\bigcup \{\sigma \,\colon (\exists C)(\langle \sigma ,C\rangle \in G)\}$ is a closed unbounded subset of S almost contained in each club set in V. $\aleph _{1}$ is preserved. This method was introduced by Ronald Jensen in order to show the consistency of the continuum hypothesis and the Suslin hypothesis.

Shooting a club with countable conditions

For S a stationary subset of $\omega _{1}$ we set P equal to the set of closed countable sequences from S. In $V[G]$, we have that $\bigcup G$ is a closed unbounded subset of S and $\aleph _{1}$ is preserved, and if CH holds then all cardinals are preserved.

Shooting a club with finite conditions

For S a stationary subset of $\omega _{1}$ we set P equal to the set of finite sets of pairs of countable ordinals, such that if $p\in P$ and $\langle \alpha ,\beta \rangle \in p$ then $\alpha \leq \beta$ and $\alpha \in S$, and whenever $\langle \alpha ,\beta \rangle$ and $\langle \gamma ,\delta \rangle$ are distinct elements of p then either $\beta <\gamma$ or $\delta <\alpha$. P is ordered by reverse inclusion. In $V[G]$, we have that $\{\alpha \,\colon (\exists \beta )(\langle \alpha ,\beta \rangle \in \bigcup G)\}$ is a closed unbounded subset of S and all cardinals are preserved.

Silver forcing

Silver forcing (after Jack Howard Silver) is the set of all those partial functions from the natural numbers into {0, 1} whose domain is coinfinite; or equivalently the set of all pairs (A, p), where A is a subset of the natural numbers with infinite complement, and p is a function from A into a fixed 2-element set. A condition q is stronger than a condition p if q extends p.

Silver forcing satisfies Fusion, the Sacks property, and is minimal with respect to reals (but not minimal).

Vopěnka forcing

Vopěnka forcing (after Petr Vopěnka) is used to generically add a set $A$ of ordinals to ${\color {blue}{\text{HOD}}}$. Define first $P'$ as the set of all non-empty ${\text{OD}}$ subsets of the power set ${\mathcal {P}}(\alpha )$ of $\alpha$, where $A\subseteq \alpha$, ordered by inclusion: $p\leq q$ iff $p\subseteq q$. Each condition $p\in P'$ can be represented by a tuple $(\beta ,\gamma ,\varphi )$ where $x\in p\Leftrightarrow V_{\beta }\models \varphi (\gamma ,x)$, for all $x\subseteq \alpha$. The translation between $p$ and its least representation is ${\text{OD}}$, and hence $P'$ is isomorphic to a poset $P\in {\text{HOD}}$ (the conditions being the minimal representations of elements of $P'$). This poset is the Vopenka forcing for subsets of $\alpha$. Defining $G_{A}$ as the set of all representations for elements $p\in P'$ such that $A\in p$, then $G_{A}$ is ${\text{HOD}}$-generic and $A\in {\text{HOD}}[G_{A}]$.

References

• Jech, Thomas (2003), Set Theory: Millennium Edition, Springer Monographs in Mathematics, Berlin, New York: Springer-Verlag, ISBN 978-3-540-44085-7
• Kunen, Kenneth (1980), Set Theory: An Introduction to Independence Proofs, Elsevier, ISBN 978-0-444-86839-8
• Kunen, Kenneth (2011), Set theory, Studies in Logic, vol. 34, London: College Publications, ISBN 978-1-84890-050-9, Zbl 1262.03001

External links

• A.Miller (2009), Forcing Tidbits.