List of forcing notions
In mathematics, forcing is a method of constructing new models M 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 ; 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 have the same cardinals.
- A subset D of P is called dense if for every there is some with.
- A filter on P is a nonempty subset F of P such that if and then, and if and then there is some with and.
- 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
Cohen forcing
In Cohen forcing P is the set of functions from a finite subset of ω2 × ω toand 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, the only restriction is that κ does not have cofinality ω.
Grigorieff forcing
Grigorieff forcing destroys a free ultrafilter on ω.Hechler forcing
Hechler forcing 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 and E is a finite subset of some fixed set G of functions from ω to ω. The element 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.
Jockusch–Soare forcing
Forcing with 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 subsets of Cantor space|, ordered by inclusion.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.- P is the set of Laver trees, ordered by inclusion.
- p is a tree: p contains any initial sequence of any element of p
- p has a stem: a maximal node such that or for all t in p,
- If and then t has an infinite number of immediate successors tn in p for.
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.- 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. 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 with domain of size less than κ and for every in the domain of p. This poset collapses all cardinals less than λ onto κ, but keeps λ as the successor to κ.
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
- P is the set of all trees which have the property that any s in T has an extension in T which has immediate successors. P is ordered by inclusion. The intersection of all trees in the generic filter defines a countable sequence which is cofinal in ω2.
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.In Prikry forcing 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.
- Finite products: If P and Q are posets, the product poset has the partial order defined by if and.
- Infinite products: The product of a set of posets, each with a largest element 1 is the set of functions p on I with and such that for all but a finite number of i. The order is given by if for all i.
- The Easton product of a set of posets, where I is a set of cardinals is the set of functions p on I with and such that for every regular cardinal γ the number of elements α of γ with is less than γ.
Radin forcing
If λ is a sufficiently large cardinal, then the forcing keeps λ regular, measurable, supercompact, etc.
Random forcing
- P is the set of Borel subsets of 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 xG in all rational intervals such that is in G. This real is "random" in the sense that if X is any subset of of measure 1, lying in V, then.
Sacks forcing
- P is the set of all perfect trees contained in the set of finite sequences. 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.
Shooting a fast club
For S a stationary subset of we set is a closed sequence from S and C is a closed unbounded subset of, ordered by iff end-extends and and. In, we have that is a closed unbounded subset of S almost contained in each club set in V. is preserved.Shooting a club with countable conditions
For S a stationary subset of we set P equal to the set of closed countable sequences from S. In, we have that is a closed unbounded subset of S and is preserved, and if CH holds then all cardinals are preserved.Shooting a club with finite conditions
For S a stationary subset of we set P equal to the set of finite sets of pairs of countable ordinals, such that if and then and, and whenever and are distinct elements of p then either or. P is ordered by reverse inclusion. In, we have that is a closed unbounded subset of S and all cardinals are preserved.Silver forcing
Silver forcing 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.
Vopěnka forcing
Vopěnka forcing is used to generically add a set of ordinals to Ordinal definable set|.Define first as the set of all non-empty subsets of the power set of, where, ordered by inclusion: iff.
Each condition can be represented by a tuple
where, for all.
The translation between and its least representation is, and hence
is isomorphic to a poset . This poset is the Vopenka forcing for subsets of.
Defining as the set of all representations for elements such that
, then is -generic and.