## 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.

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