In Section 6 of Sh410, Shelah defines several related cardinal invariants, one of which makes an appearance in a proof of the revised GCH (the one in the Abraham-Magidor chapter of the handbook). I want to use this space to clear up some of the definitions.
Def: Let be an ideal on some set . Then is -based if, whenever is such that , we also have .
Our working assumptions for this post here are the following:
- is an ideal on a cardinal .
- is not the union of countably-many sets from .
- is -based, where .
- is a cardinal with .
Def: Say a cardinal is representable if there is a collection of finite subsets of such that, for any , we have .
Def: Say a cardinal is weakly representable if there is a collection of finite subsets of such that, for any , we have .
It’s clear here that simply because the supremum is being taken over more cardinals. It turns out that these cardinals are actually equal to each other, but now that I look at the proof, I can’t really make heads or tails of it. For now, I’ll go ahead and assume that the proof is correct and see if I can piece together what’s going on later.
Anyway, we have another cardinal invariant which appears, the definition of which I want to take some time to consider. Basically, the definition is probably incorrect, given the proof of theorem 6.1, so I want to repair it in a way that makes sense.
Def: (incorrect version)
This definition looks pretty horrible, but let’s take a look at what’s going on. First, let’s replace with its counterpart to make things easier. Now, recall here that
Then, note that if then certainly and so we have that the sequence of ideals is decreasing. Now, let’s suppose that is representable by , and fix a representation of . Then for any which is positive with respect to , we have that . Now, since is larger (hance has fewer positive sets), it follows that must be representable by . In other words:
Hence, that sup is achieved by . So, as written that definition seems somewhat strange. On the other hand, what would make sense is asking for a min instead of a sup, since we get a decreasing sequence of ordinals. Finally, my advisor, Todd Eisworth, pointed out that the proof of Theorem 6.1 doesn’t actually go through as written and requires that first min actually be a sup. Now, looking at the proof, it remains intact if we actually take the definition of to be:
Def: (correct version) .
This definition actually makes much more sense, and it makes the proof of theorem 6.1 go through. So, my suspicion is that the above is the correct definition of .