List of forcing notions

From Wikipedia, the free encyclopedia

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

[edit] 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.

[edit] 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.

[edit] 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.

[edit] 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 (s,E) where s is a finite sequence of natural numbers (considered as functions from a finite ordinal to ω) and E is an element 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 t but not of s then s(k)>h(k) for all h in E.

[edit] Jockusch-Soare forcing (a.k.a. forcing with \Pi^0_1 classes)

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 2ω (meaning the sets of paths through infinite, computable subtrees of 2 < ω), ordered by inclusion.

[edit] Iterated forcing

[edit] Laver forcing

Laver forcing was used by Laver to show that Borel's conjecture 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
  • p has a stem: a maximal node 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 t'n in p for n∈ω.

[edit] 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 (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 subsetsof κ× λ 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.

[edit] Magidor forcing

Magidor forcing is a generalization of Prikry forcing used to change the cofinality of a cardinal to a given smaller regular cardinal.

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

[edit] Namba forcing

Namba forcing is used to change the cofinality of ω2 to ω.

  • P is the set of perfect trees in the set of finite sequences of ordinals less than ω2. P is ordered by inclusion.

[edit] Prikry forcing

In Prikry forcing, named after Karel Prikry, 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.

[edit] 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 Pi, 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 of a set of posets Pi, 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.

[edit] Radin forcing

Radin forcing is a generalization of Magidor forcing, used to change the cofinality of a large cardinal λ to a smaller regular cardinal κ while preserving properties of κ such as regularity.

[edit] Random forcing

  • 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 xG 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 xG ∈ X.

[edit] 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 tree s containing it 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 Sacks to produce a real a with minimal degree of constructibility.

[edit] References