## Conclusion 3.2A (part 1)

June 16, 2011 at 13:28 | Posted in Uncategorized | Leave a commentIn this post, we want to look at the next result in The Book: Conclusion 3.2A on page 378. What I will do is state this exactly as it appears in the book, and then reformulate it in conjunction with providing some commentary. I’ll attempt to prove it in a future post.

Theorem 1 (Conclusion 3.2A page 378)If , a limit ordinal, then:

- for large enough,
- if then for large enough

I’m going to concentrate on the first part of the above; here’s my attempt at reformulating it.

Theorem 2 (Conclusion 3.2A (i), reformulated)Suppose is a singular cardinal of cofinality with . Then for some ,

The right-hand side of the above is not something we’ve explicitly defined yet, but it’s a reasonable variant of the notation we’ve been working with. The notation

refers to the minimum cardinality of a family with the property that for any of cardinality , the collection

is cofinal in .

This definition is robust under small modifications: it is also the minimum cardinality of a family with the property that for any of cardinality *at most* and any , there is a finite set such that

In other words,

and we are back with the sorts of cardinals we’ve been considering of late.

There is something else implicit in this theorem that I want to bring out: it shows us that if is a singular cardinal that isn’t a fixed point, then can be calculated just by looking at structure

In this situation is the minimum cardinality of a family of functions in with the property that for some , for any of cardinality the family

is cofinal in . Roughly speaking, if is singular and not a fixed point, then is the minimum cardinality of a family of functions in which can cover all products of size drawn from some fixed tail of . That seems like it is worth knowing…

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