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.
Amoeba forcing is forcing with the amoeba order, and adds a measure 1 set of random reals.
In Cohen forcing (named after Paul Cohen) P is the set of functions from a finite subset of ω2 × ω to and if .
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 (after Serge Grigorieff) destroys a free ultrafilter on ω.
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 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 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 for all h in F.
Forcing with
| 0 | |
| \Pi | |
| 1 |
| 0 | |
| \Pi | |
| 1 |
2\omega
2<\omega
See 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 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.)
A Laver tree p is a subset of the finite sequences of natural numbers such that
If G is generic for, then the real, called a Laver-real, uniquely determines G.
Laver forcing satisfies the Laver property.
See main article: Collapsing algebra. These posets will collapse various cardinals, in other words force them to be equal in size to smaller cardinals.
Levy collapsing is named for Azriel Levy.
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.
is stronger than if s is an initial segment of t, B is a subset of A, and t is contained in .
Mathias forcing is named for Adrian Mathias.
Namba forcing (after Kanji Namba) is used to change the cofinality of ω2 to ω without collapsing ω1.
T\subseteq
| <\omega | |
| \omega | |
| 2 |
\aleph2
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
\aleph2
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.[1] [2]
In Prikry forcing (after Karel Prikrý) P is the set of pairs 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 is stronger than if t is an initial segment of s, A is contained in B, and s is contained in . This forcing notion can be used to change to cofinality of κ while preserving all cardinals.
Taking a product of forcing conditions is a way of simultaneously forcing all the conditions.
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.
See main article: random algebra.
Sacks forcing has the Sacks property.
For S a stationary subset of
\omega1
P=\{\langle\sigma,C\rangle\colon\sigma
\omega1\}
\langle\sigma',C'\rangle\leq\langle\sigma,C\rangle
\sigma'
\sigma
C'\subseteqC
\sigma'\subseteq\sigma\cupC
V[G]
cup\{\sigma\colon(\existsC)(\langle\sigma,C\rangle\inG)\}
\aleph1
For S a stationary subset of
\omega1
V[G]
cupG
\aleph1
For S a stationary subset of
\omega1
p\inP
\langle\alpha,\beta\rangle\inp
\alpha\leq\beta
\alpha\inS
\langle\alpha,\beta\rangle
\langle\gamma,\delta\rangle
\beta<\gamma
\delta<\alpha
V[G]
\{\alpha\colon(\exists\beta)(\langle\alpha,\beta\rangle\incupG)\}
Silver forcing (after Jack Howard Silver) is the set of all those partial functions from the natural numbers into whose domain is coinfinite; or equivalently the set of all pairs, 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 (after Petr Vopěnka) is used to generically add a set
A
{\color{blue}HOD
P'
OD
l{P}(\alpha)
\alpha
A\subseteq\alpha
p\leqq
p\subseteqq
p\inP'
(\beta,\gamma,\varphi)
x\inp\LeftrightarrowV\beta\models\varphi(\gamma,x)
x\subseteq\alpha
p
OD
P'
P\inHOD
P'
\alpha
GA
p\inP'
A\inp
GA
HOD
A\inHOD[GA]