On ADSFebruary 6, 2012 at 11:13 | Posted in Uncategorized | Leave a comment
So, I think I’ve finished my “busy period”, and I have a little time to write. What I want to do today is post about the topic of my second lecture at this years Czech Winter School. The very first post I made on this blog dealt with an argument of Shelah that can be used to prove Solovay’s theorem that the Singular Cardinals Hypothesis holds above a strongly compact cardinal. The argument used a consequence of as a “black box”; I want to take a look at the interior of this black box today.
Definition 1 Let a cardinal. We say that holds if there is a sequence such that each is unbounded in , and for each , the collection is essentially disjoint, in the sense that there is a function such that the collection is disjoint.
Some things to note:
- always holds if is regular (any “almost disjoint” family of size has the required property).
- If is singular and holds, then we may assume each is of order-type .
- for singular implies that that there is no countably complete uniform ultrafilter on — this is the content of the very first post I made on this blog.
- follows from the existence of a better scale (and hence it follows from each of weak square and ). The details of this are in my Handbook of Set Theory article.
What I want to do here is derive from the assumption that is a singular cardinal of countable cofinality satisfying . Remember this means that for any family of cardinality , there is a countable not covered by any member of . I’ll just state the theorem, and spread the proof out over the next couple of days in some more blog posts.
Theorem 2 (Shelah) Suppose is singular and . If , then holds.