forcing
Forcing is the method used by Paul Cohen to prove the independence of
the continuum hypothesis
(CH). In fact, the method was used by Cohen to
prove 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 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
.
We can rephrase this and say that “it is consistent with standard
field theory that have a square root”.
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.
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 |