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

Definitions

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 \supseteq 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, 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 \Pi^0_1 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^0_1 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.)

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 \in 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.

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

(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 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 \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.[1][2]

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 \cup 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.

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

Sacks forcing

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.

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

References

  1. Shelah, S., Proper and Improper Forcing (Claim XI.4.2), Springer, 1998
  2. Schlindwein, C., Shelah's work on non-semiproper iterations, I, Archive for Mathematical Logic, vol. 47, no. 6, pp. 579 -- 606 (2008)

External links

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