forcing
Forcing is the method used by Paul Cohen to prove the independence ofthe continuum hypothesis
(CH). In fact, the method was used by Cohen toprove that CH could be violated.
Adding a set to a model of set theory via forcing is similar to adjoining a new element to a field. Suppose we have a field , and we want to add to this field anelement such . We seethat we cannot simply drop a new in , since then we arenot guaranteed that we still have a field. Neither can we simplyassume that already has such an element. The standard way ofdoing this is to start by adjoining a generic
indeterminate , andimpose a constraint on , saying that . What we do is takethe quotient , and make a field out of it by taking thequotient field. We then obtain , where is theequivalence class
of in the quotient.The general case of this is the theorem of algebra saying that everypolynomial
over a field has a root in some extension field
.
We can rephrase this and say that “it is consistent with standardfield theory that have a square root”.
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 be a transitive model of set theory, which wecall the ground model. We want to “add a new set” to insuch a way that the extension
has as a subclass, and theproperties of are preserved, and .
The first step is to “approximate” the new set using elements of. This is the analogue of finding the irreducible polynomial inthe algebraic example. The set of such “approximations” can beordered by how much information the approximations give : let , then if and only if “is stronger than” . Wecall this set a set of forcing conditions. Furthermore, it is required that the set itself and the order relation be elements of .
Since is a partial order, some of its subsets have interestingproperties. Consider as a topological space
with the ordertopology. A subset is dense in if and onlyif for every , there is such that . Afilter in is said to be -generic if and only if it intersectsevery one of the dense subsets of which are in . An -genericfilter in is also referred to as a generic set of conditionsin the literature. In general, even though is a set in , generic filters are not elements of .
If is a set of forcing conditions, and is a generic set ofconditions in , all in the ground model , then we define to be the least model of ZFC that contains . Thebig theorem is this :
Theorem. is a model of ZFC, and has the same ordinals as , and.
The way to prove that we can violate CH using a generic extension isto add many new “subsets of ” in the following way : let be a transitive model of ZFC, and let be the set (in) of all functions whose domain is a finite subset of, and whose range is the set . Theordering here is if and only if . Let be a generic set of conditions in . Then is atotal function
whose domain is , and range is. We can see this as coding new functions, ,which are subsets of omega. Thesefunctions are all distinct. http://planetmath.org/node/3242doesn’t collapse cardinals since it satisfies the countable chain condition. Thus and CH is false in .
All this relies on a proper definition of the satisfaction relation in, 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 |