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 an element such . We see that we cannot simply drop a new in , since then we are not guaranteed that we still have a field. Neither can we simply assume that already has such an element. The standard way of doing this is to start by adjoining a generic indeterminate , and impose a constraint on , saying that . What we do is take the quotient , and make a field out of it by taking the quotient field. We then obtain , where is the equivalence class of in the quotient. The general case of this is the theorem of algebra saying that every polynomial over a field has a root in some extension field.
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 be a transitive model of set theory, which we call the ground model. We want to “add a new set” to in such a way that the extension has as a subclass, and the properties 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 in the algebraic example. The set of such “approximations” can be ordered by how much information the approximations give : let , then if and only if “is stronger than” . We call 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 interesting properties. Consider as a topological space with the order topology. A subset is dense in if and only if for every , there is such that . A filter in is said to be -generic if and only if it intersects every one of the dense subsets of which are in . An -generic filter in is also referred to as a generic set of conditions in 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 of conditions in , all in the ground model , then we define to be the least model of ZFC that contains . The big 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 is to 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 . The ordering here is if and only if . Let be a generic set of conditions in . Then is a total function whose domain is , and range is . We can see this as coding new functions , , which are subsets of omega. These functions 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.
|Date of creation||2013-03-22 12:44:17|
|Last modified on||2013-03-22 12:44:17|
|Last modified by||ratboy (4018)|