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单词 Forcing
释义

forcing


ForcingMathworldPlanetmath is the method used by Paul Cohen to prove the independence ofthe continuum hypothesisMathworldPlanetmath (CH). In fact, the method was used by Cohen toprove 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 anelement α such α2=-1. We seethat we cannot simply drop a new α in k, since then we arenot guaranteed that we still have a field. Neither can we simplyassume that k already has such an element. The standard way ofdoing this is to start by adjoining a genericPlanetmathPlanetmathPlanetmath indeterminate X, andimpose a constraint on X, saying that X2+1=0. What we do is takethe quotient k[X]/(X2+1), and make a field out of it by taking thequotient field. We then obtain k(α), where α is theequivalence classMathworldPlanetmathPlanetmath of X in the quotient.The general case of this is the theorem of algebra saying that everypolynomialPlanetmathPlanetmath p over a field k has a root in some extension fieldMathworldPlanetmath.

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

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

The first step is to “approximate” the new set using elements of𝔐. This is the analogue of finding the irreducible polynomial inthe algebraic example. The set P of such “approximations” can beordered by how much information the approximations give : let p,qP, then pq if and only if p “is stronger than” q. Wecall 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 interestingproperties. Consider P as a topological spaceMathworldPlanetmath with the ordertopology. A subset DP is dense in P if and onlyif for every pP, there is dD such that dp. Afilter in P is said to be 𝔐-generic if and only if it intersectsevery one of the dense subsets of P which are in 𝔐. An 𝔐-genericfilter in P is also referred to as a generic set of conditionsin 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 ofconditions in P, all in the ground model 𝔐, then we define𝔐[G] to be the least model of ZFC that contains G. Thebig theorem is this :

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

The way to prove that we can violate CH using a generic extension isto add many new “subsets of ω” in the following way : let𝔐 be a transitive model of ZFC, and let (P,) be the set (in𝔐) of all functionsMathworldPlanetmath f whose domain is a finite subset of2×0, and whose range is the set {0,1}. Theordering here is pq if and only if pq. LetG be a generic set of conditions in P. Then G is atotal functionMathworldPlanetmath whose domain is 2×0, and range is{0,1}. We can see this f as coding 2 new functionsfα:0{0,1}, α<2,which are subsets of omega. Thesefunctions are all distinct. (P,) http://planetmath.org/node/3242doesn’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 in𝔐[G], and the forcing relation. Details can be found in Thomas Jech’s book Set Theory.

Titleforcing
Canonical nameForcing
Date of creation2013-03-22 12:44:17
Last modified on2013-03-22 12:44:17
Ownerratboy (4018)
Last modified byratboy (4018)
Numerical id12
Authorratboy (4018)
Entry typeDefinition
Classificationmsc 03E50
Classificationmsc 03E35
Classificationmsc 03E40
Related topicForcingRelation
Related topicCompositionOfForcingNotions
Related topicEquivalenceOfForcingNotions
Related topicFieldAdjunction
Definesforcing
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更新时间:2025/5/4 14:44:00