## Back to the task

February 11, 2011 at 21:15 | Posted in Uncategorized | Leave a commentReturning now to our main project, we extract a lemma from Shelah’s argument and examine it in isolation.

Let and be as assumed, and consider the structure

where denotes the ideal of bounded subsets of . It’s pretty easy to see that this structure is -directed, i.e., that every collection of fewer than functions in has a -upper bound.

Thus, we can define a sequence of functions in by letting be the -least function in serving as a -upper bound for . Using this definition, we achieve

for each , and so

as well.

Lemma 1Let be a set of regular cardinals satisfying

- ,
- is progressive, and
- .

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

*Proof:* Let satisfy the assumptions of the proposition. Since , we know that carries a scale of length modulo the ideal . Since is an elementary model of and , it follows that there is such a scale living in , say . Note that each is in as .

Now consider . Since is unbounded in , it follows easily that this sequence is increasing modulo in . Given , there is an with . But clearly this implies

Thus, the sequence is also cofinal in modulo , and we are done.

Notice that the above lemma gives us a way to smuggle information into : given as in the proposition, it follows that ANY function , whether in or not, is bounded modulo by a function of the form from . Formulating this as a corollary, we obtain:

Corollary 2Let be a set of regular cardinals as in the preceding lemma. If , then there is an such that

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