## Some terminology

June 15, 2011 at 10:20 | Posted in Uncategorized | Leave a commentHaving set the mood with our last post, we now return to “The Book” and examine “covering all small products”. I want to use this post to try out some terminology to see if it’s possible to make the material a little more palatable.

Definition 1Let be a set of regular cardinals, let be a regular cardinal, and let . Given a set , we say that is -cofinal in if for any function , there is a family of fewer than elements of such that

In the last post, we characterized the minimum cardinality of a -cofinal subset of in the case where is progressive and . Today, I want to consider a related question:

What is the minimum cardinality of a family such that is -cofinal in for every of cardinality ?

Let us agree to denote the above cardinal by (again, don’t blame me for the notation!).

Ok, so why on earth would I want to consider such a cardinal? Well, we’re going to see that this is actually quite relevant for cardinal arithmetic — we will show that such cardinals are intimately tied up with Shelah’s covering numbers. Our first task, though, will be to characterize such cardinals in terms of pcf theory. The characterization will require a little terminology.

Definition 2Suppose is a progressive set of regular cardinals, and let and be cardinals such thatWe define the set to consist of all (regular) cardinals for which there is an ideal on such that

- is -complete,
- has true cofinality , and
- there is a such that and .

Notice that the above is very close to the definition of ; the difference is that the last clause further restricts the acceptable ideals to those which also “reduce” to something of the form where .

We’ll end this post with the statement of a theorem; we’ll prove the theorem in the next post.

Theorem 3Suppose is a progressive set of regular cardinals, andThen

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