## On ADS (Part 3)

February 9, 2012 at 11:40 | Posted in Uncategorized | Leave a commentContinuing the construction, let us assume that is singular of countable cofinality and . For each , let be a sequence of sets such that

- ,
- ,
- for all , and
- .

By induction on we choose sets such that for no and is a subset of .

Notice that each set of the form is of cardinality less than , and there are sets of this form. Our assumption that is greater than tells us that a suitable can always be found.

Let be a bijection, and let . We will show that the collection will witness .

What this amounts to is that for each , we need a function so that

for all , where

Lemma 1For each , the family has a transversal (i.e., a one-to-one choice function) .

*Proof:* We define by induction on . This is trivial for and limit. If , then is not a subset of so can be defined easily.

Armed with the preceding lemma, we define a function as follows:

Given , we know for some unique . We define to be the unique such that .

In English, answers the question “when does enumerate the value of ?”.

Our goal is to show that the function almost works, and then show how to modify to a function that actually does the job.

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