The Story of Generators – 3 (Existence)October 2, 2012 at 15:32 | Posted in Uncategorized | Leave a comment
The Exact Upper Bound Argument
Let us assume that is as usual, with . If we want to produce a generator for , we saw last time that what we need is a universal sequence for that in addition possesses an exact upper bound modulo .
I don’t want to make an excursion into the theory of exact upper bounds, as there have been many high quality write-ups of this material already: in addition to Section 2.1 of the Abraham/Magidor article, there are also some materials due to Kojman that do an excellent job of exposing this material. The point is that one can use results of Shelah to modify the universal sequence so that it ends up with an exact upper bound, and then we get generators by the previous post.
Instead of re-hashing such arguments, I offer the following direct route from universal sequences to generators. The proof takes advantage of the ideal , and we will see similar arguments shortly when we talk about transitivity of generators. The argument is based on the exposition of generators given in the Burke-Magidor pcf paper. It turns out that framing things in terms of simplifies things considerably.
From universal sequences to generators
Theorem 1 Suppose is a progressive set of regular cardinals and . Then there is a generator for .
Proof: We can assume that . (If either or are in , then they have generators consisting of singletons.)
Let be a universal sequence for , that is, a sequence such that
- , and
- is cofinal in whenever is an ultrafilter on satisfying .
Such sequences exist by Theorem 4.2 of the Abraham/Magidor article, or see Lemma 2.1 on page 327 of Cardinal Arithmetic.
Let . Since we have assumed , we know there is a stationary set lying in the ideal . (See the first section of [Sh:420], or Theorem 3.18 in my own Handbook article.)
This means that there is a family and a club such that
- is a closed (possibly bounded) subset of
- if then , and
- if then is club in of order-type .
For each with , we define
If , we set .
Since is -directed, we can find a single function bounding the collection modulo .
Now we let be an elementary submodel of for some sufficiently large regular cardinal such that
- , , , , , , and are all in
- is an ordinal
This is possible because is stationary in .
We prove that is a generator for . Part of this is simple, as by choice of we know . To finish, we must establish the following:
Proposition 2 If is an ultrafilter on with , then .
The proof is not difficult, but the following lemma is critical.
Lemma 3 .
This is where the assumption pays dividends. Note that as , and therefore we know that is club in with order-type .
Given in , we know that is an initial segment of , and both are initial segments of . Thus
Given and , we know there is a with . Since is cofinal in , we know is an ordinal in and . Thus , and we see
In particular, if , then there is an such that
Since is club in with order-type , we can find a single such that for all ,
The set on the right is definable from parameters available in (both and are there), and therefore .
Let us return now to the proof of Proposition 2.
Proof: By way of contradiction, suppose is an ultrafilter on forming a counterexample. Since is in , we can assume that is in as well (this is the key point).
This sequence is universal for and therefore the collection is cofinal in . Since , we know furthermore that the sequence is increasing.
Thus, there is an such that whenever . Again, we may assume that is in (and hence less than ).
Choose greater than and let . Then and . In particular, and
So we have
One the other hand, since we know
But now we have obtained a contradiction.
Proposition 2 taken together with the fact that easily establishes that is a generator for , so we are done.