This post follows up the previous one.
As before, we will let be a singular cardinal, a sequence of regular cardinals with limit , and be a fixed ideal over such that . Under these assumptions, we’ve shown that, letting
for every ultrafilter over disjoint from , if , then
If , and is proper, then is -directed. This was lemma 1.5 in Sh506. Our goal now is to use this to prove that universal cofinal sequences exist, which gives us (modulo a claim from Sh506) what Shelah calls the weak pcf theorem. Let’s take a little bit to derive some corollaries from -directedness (these appear in Sh506 as well). First note that the lemma holds trivially when since , as well as when since then has one lement. Recall that
is an ultrafilter over disjoint from . (note that every is regular)
Corollary: Under our current assumptions, for every ultrafilter over , if and only if .
Proof: By definition, implies that so this direction is uninteresting. For the other direction, suppose that , then is directed and so .
This corollary tells us that if and only if has non-empty intersection with , but misses . So if and only if is the first cardinal for which has nontrivial intersection with . It follows that we can associate to each a set , which gives us an injection from to . The idea then is that, if we can get some control over how we generate from , we should be able to say something more about the size of . Also, note that each of these sets is -positive.
We also have that exists. To see this, let
, and note that is a proper ideal since it is the union of an ascending chain of proper ideals (as implies that is proper). So, let be an ultrafilter disjoint from and let . As for all , it follows that but for all . Hence, .
Going back to the pcf ideals, we need a definition from Sh506:
Definition: We say that is semi-normal if there are for such that:
- If , then , and
Here the statement in 2. simply means that is generated from and . We also need a definition from the Abraham-Magidor chapter in the handbook.
Definition: Let be a -increasing sequence in . We say that is a universal cofinal sequence for . if, for any ultrafilter over , if , then the sequence is cofinal in .
We will take the following for granted:
Fact 2.2(2) of Sh506: is semi-normal if and only if there is a universal cofinal sequence for .
The following is buried in Sh506 (probably in the proof of Lemma 2.6), but we will instead mimic the proof of lemma 1.5 from Sh506. The proof here is also very similar to the proof that universal cofinal sequences exist given in the Abraham-Magidor chapter (except that one is for the original case).
Theorem: Every has a universal cofinal sequence.
Before proceeding, we should note that in the proof of Lemma 1.5, we actually made use of the following fact (really we kind of proved it):
is -weakly saturated if and only if every -increasing -sequence of -positive sets is eventually constant modulo .
The proof of this will be very similar to the proof of Lemma 1.5, insofar as we will proceed by induction on , and suppose that we fail to get a universal cofinal sequence at each stage. From this we will be able to produce a contradiction to weak saturation. We begin by noting that, if , then we can define by setting which is an everywhere increasing sequence. So, it trivially is a universal sequence. In that vein, we may assume that .
We will proceed by induction on , and construct candidate universal sequences . Now, we want to (as in the proof of Lemma 1.5 from Sh506) come up with sets that are -increasing in the coordinate but differ from each other modulo (and hence ). So we will ask that not only is the collection strictly increasing modulo in the coordinate, but that it is -increasing in the coordinate. With that in mind, we will use -directedness to inductively construct these sequences.
For , we let be any -increasing sequence in . We can create such a sequence inductively as follows: let be arbitrary, and then assume that have been defined for . By -directedness, we can find such that for all , and let .
At limit stages, let and assume that has been defined for each . We inductively define as follows: let , which is in since . Now suppose that has been defined for each , and let . Again , and let be such that for all by -directedness. Then define by , which is as desired.
At successor stages suppose that has been defined. If is a universal sequence, then we can terminate the induction. If not, we inductively define as follows: Since is not universal, we can find an ultrafilter over with the property that , but is -dominated by some (note that is disjoint from ). Let be a cofinal sequence in . We define by setting . Now suppose that has been defined for each , and let be such that for all by -directedness. Then define by , which is as desired. Note that is cofinal in
We claim that we must have terminated the induction at some stage. Otherwise, we will have defined for each the following:
- Sequences which are -\increasing in the coordinate, and -increasing in the coordinate.
- Ultrafilters disjoing from such that is dominated by , and is cofinal in .
We will use this to derive a contradiction. We begin by letting be defined by setting (recall that ). By condition 2 above, for every , there exists an index such that . Since for regular, it follows that is below . So, for each , we have that . Now define the sets
By construction, we have that since . On the other hand, since . So, it follows that modulo (hence modulo ). But since , we have that (in fact implies that ) and so we are in the same position as the proof of Lemma 1.5. That is, is a collection of -positive sets which are disjoint, contradiction weak saturation. Therefore, the induction must have halted at some stage and we are done.
Therefore, we have the following result:
Under our current assumptions, each is semi-normal. So, we have what Shelah refers to as the weak-pcf theorem. Our next goal is to show that this can be improved to full normality (i.e. the pcf theorem) under stronger hypotheses for our ideal .