ForcingMathworldPlanetmath is the method used by Paul Cohen to prove the independence of the continuum hypothesisMathworldPlanetmath (CH). In fact, the method was used by Cohen to prove that CH could be violated.

Adding a set to a model of set theoryMathworldPlanetmath via forcing is similar to adjoining a new element to a field. Suppose we have a field k, and we want to add to this field an element α such α2=-1. We see that we cannot simply drop a new α in k, since then we are not guaranteed that we still have a field. Neither can we simply assume that k already has such an element. The standard way of doing this is to start by adjoining a genericPlanetmathPlanetmathPlanetmath indeterminate X, and impose a constraint on X, saying that X2+1=0. What we do is take the quotient k[X]/(X2+1), and make a field out of it by taking the quotient field. We then obtain k(α), where α is the equivalence classMathworldPlanetmathPlanetmath of X in the quotient. The general case of this is the theorem of algebra saying that every polynomialPlanetmathPlanetmath p over a field k has a root in some extension fieldMathworldPlanetmath.

We can rephrase this and say that “it is consistent with standard field theory that -1 have a square rootMathworldPlanetmath”.

When the theory we consider is ZFC, we run in exactly the same problem : we can’t just add a “new” set and pretend it has the required properties, because then we may violate something else, like foundation. Let 𝔐 be a transitiveMathworldPlanetmathPlanetmathPlanetmathPlanetmath model of set theory, which we call the ground model. We want to “add a new set” S to 𝔐 in such a way that the extensionPlanetmathPlanetmath 𝔐 has 𝔐 as a subclass, and the properties of 𝔐 are preserved, and S𝔐.

The first step is to “approximate” the new set using elements of 𝔐. This is the analogue of finding the irreducible polynomial in the algebraic example. The set P of such “approximations” can be ordered by how much information the approximations give : let p,qP, then pq if and only if p “is stronger than” q. We call this set a set of forcing conditions. Furthermore, it is required that the set P itself and the order relation be elements of 𝔐.

Since P is a partial orderMathworldPlanetmath, some of its subsets have interesting properties. Consider P as a topological spaceMathworldPlanetmath with the order topology. A subset DP is dense in P if and only if for every pP, there is dD such that dp. A filter in P is said to be 𝔐-generic if and only if it intersects every one of the dense subsets of P which are in 𝔐. An 𝔐-generic filter in P is also referred to as a generic set of conditions in the literature. In general, even though P is a set in 𝔐, generic filters are not elements of 𝔐.

If P is a set of forcing conditions, and G is a generic set of conditions in P, all in the ground model 𝔐, then we define 𝔐[G] to be the least model of ZFC that contains G. The big theorem is this :

Theorem. 𝔐[G] is a model of ZFC, and has the same ordinalsMathworldPlanetmathPlanetmath as 𝔐, and 𝔐𝔐[G].

The way to prove that we can violate CH using a generic extension is to add many new “subsets of ω” in the following way : let 𝔐 be a transitive model of ZFC, and let (P,) be the set (in 𝔐) of all functionsMathworldPlanetmath f whose domain is a finite subset of 2×0, and whose range is the set {0,1}. The ordering here is pq if and only if pq. Let G be a generic set of conditions in P. Then G is a total functionMathworldPlanetmath whose domain is 2×0, and range is {0,1}. We can see this f as coding 2 new functions fα:0{0,1}, α<2, which are subsets of omega. These functions are all distinct. (P,)’t collapse cardinals since it satisfies the countable chain condition. Thus 2𝔐[G]=2𝔐 and CH is false in 𝔐[G].

All this relies on a proper definition of the satisfaction relation in 𝔐[G], and the forcing relation. Details can be found in Thomas Jech’s book Set Theory.

Title forcing
Canonical name Forcing
Date of creation 2013-03-22 12:44:17
Last modified on 2013-03-22 12:44:17
Owner ratboy (4018)
Last modified by ratboy (4018)
Numerical id 12
Author ratboy (4018)
Entry type Definition
Classification msc 03E50
Classification msc 03E35
Classification msc 03E40
Related topic ForcingRelation
Related topic CompositionOfForcingNotions
Related topic EquivalenceOfForcingNotions
Related topic FieldAdjunction
Defines forcing