In pcf theory, we look at reduced products where is a set of regular cardinals with singular limit satisfying (such sets are referred to as progressive in the literature). Of course, if is an -fixed point, we cannot ask that is a progressive interval of regular cardinals, which is also a required hypothesis of many theorems of pcf theory. Part of the point of Sh 506 is to allow us to do pcf theory at -fixed points, which interests me because this may be applicable to the problem of finding a Jónsson successor of a singular. So, I’m going to work through that paper using this blog. We first need a few definitions.
If is an ideal over a set and is a cardinal, then we say that is -weakly saturated if cannot be partitioned into -many positive sets. We say that is the smallest cardinal for which is -weakly saturated. Throughout, will denote a fixed singular cardinal with a sequence of cardinals with , and an ideal over .
is an ultrafilter disjoint from .
if is an ultrafilter over extending with , then .
For the sake of simplicity, we will just use to refer to when and are clear from the context. One of the main results of pcf theory is the existence of generators for these pcf ideals provided that is a progressive set of regular cardinals, and instrumental in that is the fact that is -directed. It turns out that we can weaken the assumption that is progressive.
Theorem(Lemma 1.5 of Sh 506): Assume that is a sequence of regular cardinals with the property that , then if is a cardinal with proper, then is -directed.
Proof: We will inductively show that, by induction on that is directed. If , then we let be defined by . Then since each is regular, it follows that and everywhere.
By way of induction, assume we have shown, for a cardinal, that is -directed, and let of size be given. We first assume that is singular. In this case, we can write such that . Then by assumption, we can bound each by some , and then bound the set by some . We then have that modulo for each .
So assume that is regular. We begin by replacing with a -increasing sequence . We just let be a -upper bound for . By construction, if we can find a such that modulo for each , then we will be done. At this point, we will proceed by induction on and attempt to construct a -increasing sequence of candidates for bounds of . As usual, we will show that this construction must terminate at some point, or we will be able to generate a contradiction.
By induction on , we will define functions , ordinals , and sequences with the following properties:
- and for all , we have that ;
- For each , and every , we have that modulo .
The construction proceeds as follows. At stage , we simply let , and set (note that only matters when for some ordinal ). At limit stages, assume that has been defined for each , and define by setting . Note since and each is regular, that .
At successor stages, let , and suppose that has been defined. If is a -upper bound for , then we’re done and we can terminate the induction. Otherwise, note that the sequence is -increasing and so there is a minimum for which every has the property that (else if there is no such , then was indeed the desired bound). By definition, that means we can find some ultrafilter , disjoint from such that and . Thus is follows that must have a -upper bound in , say . We then define by .
Note that for each , we have that . On the other hand, our definition of gives us that since is at least everywhere. Thus, condition 3 is satisfied, as are 1 and 2 trivially by construction.
We claim that this process must have terminated at some stage. Otherwise, we let , and note that each since the induction never terminated. Next, we note that condition 1) gives us that for , we have and so . Therefore, for , we have that and so the sets and are disjoint -positive sets (since extends ). But then we have a partition of into -many disjoint -positive sets, which is a contradiction. Therefore the process terminated at some point and (hence ) has a -upper bound. This completes the induction and the proof.
A couple of observations:
Since -directedness seems to be the key ingredient in obtaining universal cofinal sequences, I suspect that we can recover such sequences for each in . Further, the proof has a lot of the flavor of Shelah’s trichotomy theorem, especially the bit where we used witnessing ultrafilters to generate upper bounds for . So, I again suspect that one can prove a variant of the Trichotomy theorem for these pseudo-progressive sets of regular cardinals when we have a sufficiently nice ideal laying about. I unfortunately haven’t had time to work through the details and write this stuff down, but hopefully I’ll get around to it in the near future.
Also, it’s relatively easy to see that we get the usual characterization that for an ultrafilter over disjoint from , we have if and only if . From there, we also get a natural correspondence between and the existence of a set . We can also use this to show that has a maximum element. So, we recover a lot of nice information about the structure of from just this lemma. On the other hand, we don’t necessarily get that must be an interval of regular cardinals since that requires what Abraham and Magidor call the “no holes argument”. The only proof of that theorem that I know heavily utilizes the theory of exact upper bounds, which would need the trichotomy theorem (or something equivalent to it).