## 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…

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