Regrouping

June 9, 2011 at 14:19 | Posted in Uncategorized | Leave a comment

Ok, I’m back from a quick trip to Lawrence, Kansas and I had some time while traveling to ponder some things. I am convinced that the proof of Lemma 3.5 on Page 381 of Cardinal Arithmetic is incorrect, and the problem occurs in the last line of the proof (page 383) where he claims something is straightforward by 3.3A (page 379).

What I intend to do is the following:

  • I will back up and prove Claims 3.2, 3.2A, 3.3, and 3.4 (pages 377-279), which are applications of the technique in question for which I’m sure the given proofs work,
  • I will then prove a weakened version of Lemma 3.5 in which I am also confident, and finally
  • I will go back and look at the original proof of the “cov vs. pp” theorem (Theorem 5.4, page 87) and see what I can see.

So, let’s get to some math. We’ll start with some notation:

Definition 1 (Part of Definition 3.1 on page 376) Suppose {\mathfrak{a}} is a set of regular cardinals. Then

\displaystyle  {\rm cf}^\sigma_{<\theta}(\prod\mathfrak{a}) \ \ \ \ \ (1)

is the minimum size of a family {F\subseteq\prod\mathfrak{a}} such that for every {\mathfrak{b}\subseteq\mathfrak{a}} of cardinality less than {\theta}, if {f\in \prod\mathfrak{b}} then there is a family {H\subseteq F} of cardinality less than {\sigma} with

\displaystyle  f(\eta)<\sup\{h(\eta):h\in H\} \ \ \ \ \ (2)

for all {\theta\in \mathfrak{b}}.

The above definition is a “functional version” of Shelah’s covering numbers. The relevant section of Cardinal Arithmetic (Section 3.1 of Chapter IX) is called “The smallest number of functions required to cover all small products”, and a little contemplation of the preceding definition tells us why: We are asking about the size of a family {F} with the property that in every “small product”, every function is “covered” by fewer than {\sigma} functions from {F}.

The results we will be examining demonstrate that these two versions of covering numbers are intimately related. Claim 3.3 on page 378 provides an example of what I mean:

Theorem 2 (Claim 3.3 page 378) If {\aleph_0<{\rm cf}\sigma\leq\sigma<\theta\leq\mu<\lambda}, then

\displaystyle  {\rm cov}(\lambda,\mu^+,\theta,\sigma)={\rm cf}^{\sigma}_{<\theta}\left(\prod((\mu,\lambda)\cap{\sf Reg})\right)+\lambda. \ \ \ \ \ (3)

Recall that the left-hand side denotes the minimal cardinality of a family {\mathcal{P}\subseteq [\lambda]^{\leq\mu}} with the property that every set in {[\lambda]^{<\theta}} can be covered by a union of fewer than {\sigma} elements of {\mathcal{P}}. Roughly speaking, the complicated part of the right-hand side asks for the minimum cardinality of a family

\displaystyle  F\subseteq\prod((\mu,\lambda)\cap{\sf Reg}) \ \ \ \ \ (4)

with the property that every partial function in the product with domain of cardinality less than {\theta} is covered by a family of fewer than {\sigma} sets from {F}. The analogy between the definitions of the two sides is clear, and the theorem asserts that the analogy has meaning.

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