## Pedagogical Remarks

September 28, 2011 at 15:21 | Posted in Uncategorized | Leave a commentIn 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 1Let be a singular cardinal, with as usual. Suppose is an elementary submodel of , is a family of functions in , and . We saysupports beyondif

- , and
- for every and , there is an such that .

If I’ve done things right, the following should be true.

Theorem 2Let 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 ,
then

*Proof:* Assume the above. Given a finite subset of , let

and define

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…

## Finishing up Claim 3.3

September 21, 2011 at 16:35 | Posted in Uncategorized | Leave a commentSo 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 1Suppose 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 .
Then

*Proof:* Suppose this fails, and define

and

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.

## Proof of Claim 3.3 continued…

September 21, 2011 at 15:05 | Posted in Uncategorized | Leave a commentIn 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 1Let be a singular cardinal, and supposeFurther 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.,

and

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

and

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.

## Ignore that last post…

September 21, 2011 at 14:25 | Posted in Uncategorized | Leave a commentActually, 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

and 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:

and

This suffices, because

and since , we will have shown that can be covered by a union of fewer than sets from .

## Chugging along

September 12, 2011 at 16:35 | Posted in Uncategorized | Leave a commentKeeping 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 .

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