First steps in the hard direction

May 10, 2011 at 16:01 | Posted in Uncategorized | Leave a comment

Now what about the “hard direction”? Given cardinals {\sigma}, {\theta}, and {\mu} such that {\sigma} is regular and {\aleph_0<\sigma\leq{\rm cf}(\mu)<\theta\leq\mu}, our aim is to prove

\displaystyle  {\rm cov}(\mu,\mu,\theta,\sigma)\leq{\rm pp}_{\Gamma(\theta,\sigma)}(\mu), \ \ \ \ \ (1)

and moreover, if {{\rm cov}(\mu,\mu,\theta,\sigma)} is regular, then there exist {\mathfrak{a}} and {I} such that

  1. {\mathfrak{a}} is a progressive set of regular cardinals with {\sup(\mathfrak{a})=\mu} and {|\mathfrak{a}|<\theta},
  2. {I} is a {\sigma}-complete ideal on {\mathfrak{a}} extending the bounded ideal,
  3. the structure {(\prod\mathfrak{a}, <_I)} has a true cofinality, and
  4. {{\rm cov}(\mu, \mu, \theta, \sigma)\leq {\rm tcf}(\prod\mathfrak{a}, <_I)}.

Notice that by the preceding post, condition 4 is equivalent to the statement

\displaystyle  {\rm cov}(\mu, \mu, \theta,\sigma)={\rm tcf}(\prod\mathfrak{a}, <_I), \ \ \ \ \ (2)

and so the equality

\displaystyle  {\rm pp}_{\Gamma(\theta,\sigma)}(\mu)={\rm cov}(\mu,\mu,\theta,\sigma) \ \ \ \ \ (3)

holds (at least in the case where {\aleph_0<\sigma}) in a strong sense. Note that the underlying issue here is that the pp number is defined as the supremum of a certain set of regular cardinals; we will show that if this supremum itself is a regular cardinal (and hence weakly inaccessible), then it actually occurs as the true cofinality of {(\prod\mathfrak{a}, <_I)} for some relevant {\mathfrak{a}} and {I}.

We’ll do this by proving the following:

Theorem 1 Suppose {\lambda} is a regular cardinal such that {{\rm tcf}(\prod\mathfrak{a}, <_I)<\lambda} whenever {\mathfrak{a}} and {I} satisfy conditions 1, 2, and 3 above. Then {{\rm cov}(\mu,\mu,\theta,\sigma)<\lambda} as well.

The usual caveats apply here: it is closer to the truth to say that I will attempt to prove the above theorem. We’ll see what happens…

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