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.
Contents
- Notation
- Definitions
- Amoeba forcing
- Cohen forcing
- Grigorieff forcing
- Hechler forcing
- JockuschSoare forcing
- Iterated forcing
- Laver forcing
- Levy collapsing
- Magidor forcing
- Mathias forcing
- Namba forcing
- Prikry forcing
- Product forcing
- Radin forcing
- Random forcing
- Sacks forcing
- Shooting a fast club
- Shooting a club with countable conditions
- Shooting a club with finite conditions
- Silver forcing
- References
Notation
Definitions
Amoeba forcing
Amoeba forcing is forcing with the amoeba order, and adds a measure 1 set of random reals.
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
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, the only restriction is that κ does not have cofinality ω.
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
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.)
A Laver tree p is a subset of the finite sequences of natural numbers such that
If G is generic for (P,≤), then the real {s(p) : p
Laver forcing satisfies the Laver property.
Levy collapsing
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.
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
Mathias forcing is named for Adrian Richard David Mathias.
Namba forcing
Namba forcing (after Kanji Namba) is used to change the cofinality of ω2 to ω without collapsing ω1.
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
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
Product forcing
Taking a product of forcing conditions is a way of simultaneously forcing all the conditions.
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
Sacks forcing
Sacks forcing has the Sacks property.
Shooting a fast club
For S a stationary subset of
Shooting a club with countable conditions
For S a stationary subset of
Shooting a club with finite conditions
For S a stationary subset of
Silver forcing
Silver forcing (after Jack Howard Silver) satisfies Fusion, the Sacks property, and is minimal with respect to reals (but not minimal).