## The main proof, part 1.

February 14, 2011 at 12:13 | Posted in Uncategorized | Leave a commentGeneral results on the approachability ideal tell us there is a sequence

such that

- is a closed (not necessarily unbounded!) subset of ,
- has order-type at most , where ,
- , and
- the set of of cofinality with is stationary.

The above is a result from [Sh:420]; it says that there is a stationary subset of in the ideal . The details of this are worked out in my Handbook article — see Theorem 3.18 and Theorem 3.7.

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Without loss of generality, the sequence lies in the model , hence in each and as well. Let be the -least continuous and increasing sequence of cardinals cofinal in satisfying ; note this means that the sequence is going to be in every elementary submodel of the structure .

Moving back to the main proof, let us fix ; recall that we want to show is a subset of some element , and the preceding post gives us our strategy.

For each and , we define

and

We note the following:

- ,
- , and
- both and have cardinality .

Next, we define

and let be the characteristic function of (note the “+”!) in , i.e., the domain of is , and for , we have

The function lies in , but there is no reason to believe it is an element of . Note, however, that IS an element of . It is also a progressive set of regular cardinals cofinal in , and therefore by earlier work, we know

is a scale in modulo the ideal of bounded subsets of .

Since is -directed modulo the ideal of bounded subsets of , it follows that for each , there exists a such that

The function mapping to need not be in . Nevertheless, the set of is closed unbounded in , and so we can find a of cofinality such that

- is closed under the map , and
- is closed unbounded in (of order-type ).

We’ll end this post on this cliff-hanger…

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