Second Project: Introduction

April 8, 2011 at 12:42 | Posted in Uncategorized | Leave a comment

One of the goals I set for myself is to use this blog to work through some of the “cov vs. pp” material in Shelah’s oeuvre, so I want next to take a look at the so-called “cov vs. pp theorem”: Theorem 5.4 on page 87 of Cardinal Arithmetic. This proof seems to date from the era before the discovery of pcf generators, so the theorem cries out for a new proof. The argument form Lemma 3.5 on page 381 of Cardinal Arithmetic looks like it might do the job, so that’s what my second project will be — I will attempt to give a reasonable proof of the following:

Theorem 1 Suppose {\aleph_0<\sigma\leq{\rm cf}(\mu)<\theta<\mu} where {\sigma} and {\theta} are regular. Then

\displaystyle  {\rm cov}(\mu,\mu,\theta,\sigma)=^+{\rm pp}_{\Gamma(\theta,\sigma)}(\mu). \ \ \ \ \ (1)

There’s a lot of notation packed into the statement of this theorem, so we’ll use the rest of this post to define everything. Roughly, the theorem pins down how cardinals determined by pcf theory correspond to some natural measures of the powerset of a singular cardinal {\mu}. This connection is well-known in the case where {\mu} is not a fixed point; the above theorem holds even if {\mu} is a fixed point, but at the cost of assuming the cofinality of {\mu} is uncountable.

Definition 2 Let {\sigma}, {\theta}, and {\mu} be as above.

  1. {{\rm cov}(\mu,\mu,\theta,\sigma)} is the minimum cardinality of a set {\mathcal{P}\subseteq[\mu]^{<\mu}} such that any element of {[\mu]^{<\theta}} can be covered by a union of fewer than {\sigma} sets from {\mathcal{P}}.
  2. {{\rm pp}_{\Gamma(\theta,\sigma)}(\mu)} is defined to be the supremum of all cardinals {\kappa} for which there is a cardinal {\tau<\theta}, a {\sigma}-complete ideal {I} on {\tau}, and a sequence {\langle \mu_i:i<\tau\rangle} of regular cardinals below {\mu} such that
    • {\{i:\mu_i<\epsilon\}\in I} for each {\epsilon<\mu}, and
    • {\kappa={\rm tcf}(\prod_{i<\tau}\mu_i, <_I)}
  3. Notice that the pp number is the supremum of a set of cardinals, while the cov number is defined to be the cardinality of a certain collection of sets. When we say these two cardinals are {=^+}, we mean that if the covering number is regular, then it is realized as the true cofinality of some product as in the definition of the pp number.

We noted earlier that the theorem assumes the singular cardinal in question has uncountable cofinality. The question of whether this hypothesis is necessary is still open; eventually, we will look at this problem in more detail.

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