On ADS

February 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 {{\rm pp}(\mu)>\mu^+} as a “black box”; I want to take a look at the interior of this black box today.

Definition 1 Let {\mu} a cardinal. We say that {{\rm ADS}_\mu} holds if there is a sequence {\langle A_\alpha:\alpha<\mu^+\rangle} such that each {A_\alpha} is unbounded in {\mu}, and for each {\beta<\mu^+}, the collection {\{A_\alpha:\alpha<\beta\}} is essentially disjoint, in the sense that there is a function {F_\beta:\beta\rightarrow\mu} such that the collection {\{A_\alpha\setminus F_\beta(\alpha):\alpha<\beta\}} is disjoint.

Some things to note:

  • {{\rm ADS_\mu}} always holds if {\mu} is regular (any “almost disjoint” family of size {\mu^+} has the required property).
  • If {\mu} is singular and {{\rm ADS}_\mu} holds, then we may assume each {A_\alpha} is of order-type {{\rm cf}(\mu)}.
  • {{\rm ADS_\mu}} for {\mu} singular implies that that there is no countably complete uniform ultrafilter on {\mu^+} — this is the content of the very first post I made on this blog.
  • {{\rm ADS_\mu}} follows from the existence of a better scale (and hence it follows from each of weak square and {{\rm pp}(\mu)>\mu^+}). The details of this are in my Handbook of Set Theory article.

What I want to do here is derive {{\rm ADS_\mu}} from the assumption that {\mu} is a singular cardinal of countable cofinality satisfying {{\rm cov}(\mu,\mu,\aleph_1, 2)>\mu^+}. Remember this means that for any family {\mathcal{F}\subseteq[\mu]^{<\mu}} of cardinality {\mu^+}, there is a countable {A\subseteq\mu} not covered by any member of {\mathcal{F}}. 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 {\mu} is singular and {{\rm cf}\mu=\aleph_0}. If {{\rm cov}(\mu,\mu,\aleph_1, 2)>\mu^+}, then {{\rm ADS}_\mu} holds.

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