I want to make good on the promise of linking up scales to square-like principles that I made a couple of posts ago. In particular I want to sketch how one produces scales using the machinery of . If we believe that contains a lot of stationary sets, then most of the work was actually done by working through Claim 2.6A of Chapter 1 from Cardinal Arithmetic:
Theorem: Let be an ideal on a set of regular cardinals with regular. Assume that:
- is regular such that there is some stationary which has a continuity condition ;
- is a sequence of functions from to the ordinals;
- obeys .
Then has a -exact upper bound.
We just have to show that it’s we can construct sequences which obey continuity conditions, and then there’s a relatively standard argument which allows us to move immediately from an exact upper bound for an appropriate sequence to a scale. Let’s briefly recall what continuity conditions are, and how they relate to :
Theorem (Shelah): Let be a regular cardinal. Then for , we have if and only if there is a sequence and a club such that:
- Each is a closed (but not necessarily unbounded) subset of ;
- if then ;
- If , then is singular, and is a club subset of of order type .
The sequence is called a continuity condition for , and functions somewhat like a square sequence over . The major difference is that we only have coherency on the non-accumulation points which is a significant weakening, but still allows them to be useful enough. So the fact that contains a stationary subset of for every regular such that can be regarded as a weak fragment of square which is true in ZFC.
Since we want to produce scales, our focus will be on for singular. In particular, we have that for every regular , there is a stationary such that . Further, we also have that if is strong limit, then , though this won’t be particularly important to us (a proof of this can be found in Todd Eisworth’s Handbook chapter for the interested).
First, we show how to produce sequences that obey continuity conditions. So, fix a set of regular cardinals cofinal in with and such that . Following standard notation, we will let denote the ideal of bounded subsets of .
We first show that is directed. To see this, note that is suffices to show that is -directed. For any set such that , then we can rewrite where . From there, we use -directedness to produce bounds for each , and then bound by . For -directedness, let be such that . Then let be defined by , and note that is defined almost everywhere since each is regular and is cofinal in . Clearly then is a -upper bound for .
Now let be regular, and let be such that with a witnessing continuity condition. We first recall what it means for a sequence to obey :
Definition: We say weakly obeys if:
If is such that , then for each , we have for each .
This definition looks like a weakening of the one originally given, but it’s all that was required for the proof of Claim 2.6A to go through. Now we inductively define a sequence which obeys as follows. We first let be any function in . At stage , we suppose that has been defined for each . We let be a -upper bound for as guaranteed by -directedness. If is empty or , then we just set . Otherwise, we let be defined by setting . Note that since , we know that is defined almost everywhere. It is also clear by construction that is a -increasing sequence which weakly obeys and so we are done.
I also want to note that we could have started with a fixed sequence , and asked that not only weakly obey , but also that for each . So for example if weakly obeyed some other continuity condition , then the resulting would weakly obey both and . Further, if obeys a continuity condition for a stationary subset of , then one can show that the exact upper bound produced by Claim 2.6A satisfies:
Okay, with all of this in hand, we can produce a -increasing sequence of functions in with the following properties:
- has an exact upper bound ;
- For every regular with , the set .
So, we then have that sequence witnesses that has true cofinality . Now, by possibly altering on a null set we may assume . Let , and note by condition 2, that is cofinal in and has order type . A relatively standard argument then allows us to conclude that , and letting the witnessing sequence be , we get that is a scale on .
Honestly, parts of this sketch are pretty bare-bones, but the idea was to show that Claim 2.6A (once appropriately modified) is the only really difficult part behind producing scales. In fact, that claim plays the same role that the trichotomy theorem does for the theory of exact upper bounds. In particular, it shows us that, provided we can construct certain sorts of sequences, we can then get nice exact upper bounds. It just turns out that these sequences, once we have enough of the combinatorics in hand, are relatively easy to produce. From there, it’s just standard arguments showing that we really only need exact upper bounds to do a lot of the things we want. An alternative approach to exact upper bounds (outside of or trichotomy) is also furnished through what Abraham and Magidor call . It turns out that is incredibly similar to having continuity conditions for a stationary subset of lying around.
Overall though, all three of these approaches are doing roughly the same thing.