Towards Forcing: Extensions of Structures

Last time

Unfortunately, I don’t have a result to work towards with this post. Instead, I want to give a rough idea of what forcing is, by analogy with other extensions of structures, in particular, field extensions and Cauchy completions.

The usual way to construct an extension F over a base field K is to construct F as a quotient of the polynomial ring K[x] by some (irreducible) ideal I. Abstracting a bit, the sketch of this construction is:

  • Start with a base structure M. (In this case, our base field K.)
  • Find an object which approximates some gap in our structure. (In this case, the ideal I represents an approximation of a solution to some equation (or set of equations) in K.)
  • Use some common operation to generate a new structure which contains the approximated object. (In this case, take the quotient K[x]/I.)

A similar thing happens when we complete a metric space:

  • Start with a metric space M.
  • Take the set C of Cauchy sequences in M. Here, Cauchy sequences are approximations of points which don’t exist in M.
  • Quotient C by Cauchy equivalence.

Compactification is another example of this type of construction. The basic idea of forcing is similar, but we need to figure out:

  • What is our base structure?
  • How do we approximate an object which doesn’t exist?
  • What operation do we use to finish the approximation?

The base structure is easy: we start with a model of set theory, M. For technical reasons, we need this to be a model of a “sufficiently large fragment of ZF”. What this means isn’t important, because we’ll just tacitly assume we’re always working with models of ZF (even though this isn’t quite justified.) We also want this to be a transitive model, because certain technical lemmas (that we’ll avoid) require this. Sorry, this is all a little sloppy; we’ll go into more detail next time.

What are the approximations? Forcing makes use of a poset P (called a forcing notion) which exists in M (That is, P\in M.) The elements of P are the approximations of an object which doesn’t exist in M, and we want to add to M the “limit” (which won’t exist) of a set of elements of P. (edit: I said supremum before, but that’s not really what we’re looking for.)

The subset of P we use is what’s called a generic filter. A filter in P is a subset F of P such that

  • F is non-empty;
  • F is upward closed. That is, if p\in F and p\le q, then q\in F;
  • Every two elements of F are compatible. That is, for every p,q\in F, there is some r such that r\le p and r\le q.

Aside: this is nothing more than a generalization of the notion of filter for Boolean algebras (or lattices in general). Since in a Boolean algebra, a filter is dual to an ideal, by taking a filter we are morally doing the same thing as taking an ideal in a ring.

A subset D of P is called dense if for every p\in P there is some q\in D with q\le p (that is, you can always find an element of D below any element of P, and D is called open if it is downward closed.

A filter F is called generic over M if additionally, F intersects every dense open subset of P which is an element of M.

Genericity expresses “non-existence of an object” in much the same way that non-convergence of a Cauchy sequence or irreducibility of an ideal expresses non-existence of an object.

It’s a fact (which I won’t prove), that for every poset P\in M, there is a generic filter G over M. However, we want G to not exist in M, and to ensure this, we require P to be perfect; that is, for every p\in P, there are q,r\le p which are incompatible. If P is perfect, it is in fact the case that there are generic filters over M that are not elements of M.

To complete the picture, we need to know how to generate the new structure. For this, we use relative constructability, which requires a more thorough explanation than I’m prepared to give in this post. We’ll leave a discussion of that, as well as some other important details for the next post. At any rate, when we’re finished, we’ll have a structure denoted M[G], which has M as a subset at G as an element. Usually, by letting c=\bigcup G, we have that c\not\in M, but c\in M[G]. Then c is a new element of our universe, and is the element “approximated” by G.

Before I leave off, I want to give a sketch of the first forcing argument: adding a new real number.

As our notion of forcing, we take what’s called the Cohen forcing (denoted \mathbb{C}), which is the set of all finite sequences of \{0,1\} ordered by reverse inclusion; that is, \mathbb{C} is the complete binary tree turned upside down. The idea is that the elements of \mathbb{C} represent approximations of subsets of \mathbb{N}, where as we move down the tree (away from the root) we get a better approximation. Observe that \mathbb{C} is perfect: We can extend any sequence p by adding either 1 or 0, and these two extensions will be incompatible.

A filter in \mathbb{C} is a branch in the tree (being just a filter it need not be a complete branch—it may cut off somewhere). A dense subset D is one which extends every possibly sequence. That is, given any sequence, say s=\langle 0,1,0\rangle, we have some sequence in D which extends s, maybe something like \langle 0,1,0,0,1\rangle.

Genericity ensure two things:

  • The branch is a “complete” branch–it is a path all the way from the root to “the end”.
  • The branch does not exist in M.

Once we have a generic filter G, when we construct M[G], we get an infinite sequence c=\{c_n:n\in\mathbb{N}\} which is the limit of the sequences in G. By genericity, we must have that c\not\in M. But this sequence corresponds to a real a number; if the real number existed in M, then so would c, so we know that c is a new real number. Such a new real number is called a Cohen real.

We can use this idea to construct a model of ZF in which the continuum hypothesis fails: instead of just adding one real number, we can add as many real numbers as we want.

Next time, after we discuss relative constructibility and some other minutia of forcing, we will see exactly how to “refute” the continuum hypothesis.

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