In this post, I want to take a stab at “pedagogically reorganizing” the preceding proof. This post should be considered tentative, as I’m just trying to work out what the best definitions should be.
Definition 1 Let be a singular cardinal, with as usual. Suppose is an elementary submodel of , is a family of functions in , and . We say supports beyond if
- , and
- for every and , there is an such that .
If I’ve done things right, the following should be true.
Theorem 2 Let be a singular cardinal, and assume with infinite. If has the property that for every , there is an elementary submodel of such that
- , and
- is supported past by a subset of of cardinality less than ,
Proof: Assume the above. Given a finite subset of , let
Clearly and are of the same cardinality (our hypotheses certainly imply is infinite!). Since , we will be done provided we show that every can be covered by a union of fewer than sets in .
Given , let be an elementary submodel as in the assumptions of the theorem, and let be a subset of such that
- , and
- supports beyond .
Let us define
and note that Lemma 1 in the preceding post tells us
But is the union of fewer than sets from , as
and so we’re done.
What’s the point? Well, the above emphasizes that we can bound covering numbers by the sizes of “supportive” families of functions in . This is “what’s going on” in this section of The Book, and I think the proofs Shelah gives are best viewed as ways of constructing “supportive families”. We’ll see how it goes when I look at the next result in the chapter…
So why does work? Let us define
we should show
It should come as no surprise that we’re going to accomplish this using our prior postings on elementary submodels. In particular, we need to jump back to our posting of May 24, where we proved a version of Shelah’s Claim 3.3A. The notation and hypotheses are slightly different in our current setting, and since last May is long time ago, I’ll just formulate another version of things and then prove it.
Lemma 1 Suppose is a singular cardinal and is as usual. Suppose we have objects , , and such that
- is an elementary submodel of ,
- is a cardinal less than and in ,
- , and
- for every and , there is an such that .
Proof: Suppose this fails, and define
It should be clear that is well-defined and . It is also obvious that . We will now show that .
This follows immediately if , as then will be unbounded in . If , then
and so ordinals of the form for are cofinal in . But each of these ordinals is also in (as is in together with all members of ), and this tells us that must contain an ordinal bigger than .
Thus, no matter what the cofinality of , we know
Pausing for a moment, we note that the salient property of guarantees
This is quite powerful, given that we know . For example, this immediately implies that cannot be a successor ordinal, for example, as its predecessor would violate the above condition. Similarly, must have cofinality greater than — if not, then we can use the fact that to derive a contradiction.
We’ll finish by showing that also leads to a contradiction. Given the definition of , we know that any element of belongs to a subset of of cardinality at most that is definable in from finitely many members of . In particular, there is a set such that
- , and
- is definable in from finitely many members of .
(We discussed this in more detail back in May.)
The definability of guarantees that it is an element of as well, and therefore as it can be defined as the minimal element of .
Thus is an element of . Let us fix a strictly increasing function mapping onto a cofinal subset of . Let us define
Note that . We have assumed , so this means
Thus our assumptions give us an with .
But is an element of as well, hence so is the ordinal . We’ve now got a problem: since is an increasing function from into , it follows that
contradicting the fact that .
Now that we’ve established the lemma, let’s look back at what we need to verify in order to apply it. Looking at the bullet points in the statement of our lemma, all we need to worry about is the fourth one. Suppose and . Choose large enough so that
Looking back at the construction, we then have
and our choice of guarantees the existence of a function with
Thus, the hypotheses of the lemma have been satisfied by the construction from the previous post, and we are finished.
In the last post I told you what I am going to do, and in this post I want to formulate things as a stand-alone lemma:
Lemma 1 Let be a singular cardinal, and suppose
has the property that whenever is a subset of of cardinality less than , for any there is a family of cardinality such that
Then given , there is a set of cardinality such that
where is “as usual”.
I don’t know if I’ll give the entire proof in this post, but I’ll at least get started.
So let be a subset of of cardinality less than . We now define (by recursion on ) objects and such that
- each is an elementary submodel of of cardinality less than , and
- each is a subset of of cardinality less than .
The construction starts by setting
Now assume that we are given the model . We assume that our construction has achieved and so
is of cardinality . We let be the characteristic function of in , i.e.,
for each . (Notice that this definition makes sense, as .)
By our hypothesis, there is a family of cardinality such that
We now define
and the construction continues. (Note that is still of cardinality less than , as had cardinality less than .)
In the end, let us define
At this point, our assumption that has uncountable cofinality becomes important, as this implies
In our next post, we’ll show that this does the job.
Actually, I think I lied to you in the last post, at least as far as our strategy is concerned. I’ll outline another route to the result that is closer to what Shelah does; it’s actually shorter than what I had in mind originally.
Recalling our situation, is a singular cardinal, and is as usual. We have also assumed . Let us assume that is a family of functions as required in the definition of , i.e., a collection of functions that are “-cofinal in any subproduct of size constructed from ” (see previous posts for a more precise definition, but things should be clear from the construction).
Given a finite subset of , we define
Note that each is a subset of of cardinality , so
Also, it should be clear that
so we will be finished if we can prove that any can be covered by a union of fewer than sets drawn from .
How will we do this? Given , we will produce a family satisfying the following two properties:
This suffices, because
and since , we will have shown that can be covered by a union of fewer than sets from .
Keeping the notation the same as in the last post, we begin our quest to prove . The proof is not surprising. Start by letting
be a family of cardinality “covering all small products”. We let be a sufficiently large regular cardinal, and let be an elementary submodel of containing the relevant parameters, with
Our plan is to prove that
is a family satisfying the needed “covering requirements”. Said another way, we will prove that any element of can be covered by a union of fewer than sets from . Since
this will establish .