So one of my goals for this semester (or year) is to try and figure out what’s going on in Section 2 of Sh460. Of course, the section starts off by referencing Claim 6.7A of Sh430 and improving it (without mentioning what’s actually going on in that claim). Looking back at Claim 6.7A of Sh430, it turns out that this references some of the tools used in the proof of Claim 6.7, which gives us the existence of closed and transitive generators. Now it turns out that one of the things that we worked through in the summer school at UC Irvine (which I like to call pcf-fest 2016) is this very thing.
The proof that James gave was a bit different, but I think that claim 6.7A is really just making more explicit the relationship between transitive generators, universal sequences, and -IA elementary substructures. So what I’d like to do first is go back and work through the existence of transitive generators, and see how much of this stuff I can tease out along the way. Hopefully that’ll also put me in a good mindset to work through the Sh460 stuff. I figured that a good place to start is with the usual construction of generators and how they relate to universal sequences.
Throughout this, I’m going to let be a collection of regular cardinals, and put restrictions on it as necessary.
Definition: Let be a set of regular cardinals, and define
Here is just domination modulo . I will frequently bounce between and .
Definition: Let be a regular cardinal, then
Note that this is an ideal on .
Definition: We say that is a generator of if , and is generated from by .
In particular, we see that . Also note that if , then obviously for . So in the case that , asking for a generator is fantastically uninteresting. Now, let’s say that is progressive whenever .
Definition: Let , then is a universal sequence for if:
- is -increasing;
- For any ultrafilter over such that , we have that is cofinal in .
Note that if is an ultrafilter with , then . Otherwise, there is some with and . But then, since , which would mean that . This gives us another characterization of as the collection of subsets of which forces whenever they get assigned measure one by .
Theorem (Shelah): If is progressive, then for every , there is a universal sequence for with a exact upper bound .
Why are universal sequences useful? Well, if is a universal sequence for with exact upper bound , then the set is actually a generator for . Now, these generators are only unique modulo , and so we have some room to massage them. In the next post, I want to examine the possibility of doing just that.