[Sh:430.1-1] What are we aiming for?

January 24, 2011 at 21:38 | Posted in Cardinal Arithmetic, [Sh:430] | Leave a comment

We focus our attention on the first section of [Sh:430]: “equivalence of two covering properties”. The first result states the following:

Theorem 1 If {\rm{pp}(\lambda)=\lambda^+}, {\lambda>\rm{cf}(\lambda)=\kappa>\aleph_0} then {\rm{cov}(\lambda,\lambda,\kappa^+,2)=\lambda^+}.

I am pretty sure that this isn’t what Shelah meant to write here — the assumption that {\lambda} has uncountable cofinality means that the result is a trivial consequence of a deeper theorem in Cardinal Arithmetic [Update: Perhaps not! See 2nd update below]. I think that the proof he gives works even for the countable cofinality case, and this gives us something of interest because {\lambda} could very well be a fixed point. So, here’s what I conjecture he meant to say (after changing notation and translating the “cov” statement into more standard form):

Theorem 2 Let {\mu} be a singular cardinal. Then {\rm{pp}(\mu)=\mu^+} if and only if there is a family {\mathcal{P}\subseteq [\mu]^{<\mu}} of cardinality {\mu^+} such that every member of {[\mu]^{\rm{cf}(\mu)}} is a subset of some element of {\mathcal{P}}.

We’ll work through his proof as best we can, and see if my conjecture is correct.

UPDATE 1: I don’t think the conjecture is correct. It looks to me like the proof was originally written for singular of countable cofinality, but a mistake was discovered and the statement was corrected but a lot of the old proof didn’t get revised properly. The annotated content was revised, but the abstract was not. Anyway, I’ll ask Saharon if I can’t figure out what’s going on.

UPDATE 2: I think the theorem doesn’t follow from the Cardinal Arithmetic stuff. The problem I run into is that when Shelah says informally that “cov = pp when the cardinal has uncountable cofinality”, there are lots of disclaimers hidden in the background. But this underscores why I think what I’m doing here is important — I want to pin down what exactly is known and what exactly is still open in this area.

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