## Characterize the sup of pcf sigma-complete

June 10, 2011 at 13:31 | Posted in Uncategorized | Leave a commentThis post should be considered a warmup. To start, I want to recall one of the standard results of pcf theory, namely that if is a progressive set of regular cardinal, then

where the ordering on is given by

My aim here is to prove an analogous result characterizing . For those new to this, a cardinal is in if there is a -complete ideal on with

which means that there is a -increasing -sequence of functions in cofinal in . In other words, (3) means that has cofinality , *AND* every subset of of cardinality less than has an upper bound in .

The characterization we give involves the following notion:

Definition 1Let be a regular cardinal, and suppose is a set of regular cardinals. We say that a family is -cofinal in if for every , there is a family of size less than such that , i.e., such that

for all . The -cofinality of is defined to be the minimum cardinality of a -cofinal subset of . We will denote this cardinal by .

So now we come to the promised characterization:

Theorem 2If is a progressive set of regular cardinals, and is a regular cardinal with , then

*Proof:* We prove the direction first, so assume is a -cofinal family in , and . Let be a -complete ideal on such that

and assume by way of contradiction that . Given this, (15) implies that is bounded modulo in , so there is a with

We have assumed, however, that is -cofinal in , so we can fix and with

Given , define

By our choice of , we have

Since is -complete and , there is a with . But this is a contradiction as was an upper bound for in . Thus, and the inequality “” has been established.

For the other direction, we are going to need a little pcf theory. In general, for each we have an associated ideal and generator , and

This entails the existence of a sequence of functions witnessing the true cofinality of the partial order, and we let denote the collection .

Let us define

Clearly . We claim that is -cofinal in .

Suppose this is not the case. Then there is a function that cannot be covered by fewer than functions from . We define an collection of subsets of by putting into if and only if there is a family of cardinality such that

It is easy to check that is a -complete ideal on , and this ideal is proper because of our choice of .

Let be minimal with , and define

Since is -complete, it follows easily that is as well, and our choice of guarantees that is a proper ideal on . Our choice of also guarantees

Thus, the functions show us

and therefore

We begin to close in on a contradiction. We know that is a proper ideal on and therefore (we’ll drop the “”) from now on) is not in ; we will finish the proof by demonstrating that in fact *MUST* be in .

Given the definition of , we know there is an for which

Said another way,

where the last inclusion is from (15).

But since , we know

as well, by the very definition of .

Thus, is the union of two sets in , hence an element of . Contradiction.

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