Some terminology

Having set the mood with our last post, we now return to “The Book” and examine “covering all small products”. I want to use this post to try out some terminology to see if it’s possible to make the material a little more palatable.

Definition 1 Let ${\mathfrak{a}}$ be a set of regular cardinals, let ${\sigma}$ be a regular cardinal, and let ${F\subseteq\prod\mathfrak{a}}$. Given a set ${\mathfrak{b}\subseteq\mathfrak{a}}$, we say that ${F}$ is ${<\sigma}$-cofinal in ${\prod\mathfrak{b}}$ if for any function ${f\in\prod\mathfrak{a}}$, there is a family ${F_0}$ of fewer than ${\sigma}$ elements of ${F}$ such that

$\displaystyle f(\eta)<\sup\{g(\eta):g\in F_0\}\text{ for all }\eta\in \mathfrak{b}. \ \ \ \ \ (1)$

In the last post, we characterized the minimum cardinality of a ${<\sigma}$-cofinal subset of ${\prod\mathfrak{a}}$ in the case where ${\mathfrak{a}}$ is progressive and ${\sigma<|\mathfrak{a}|}$. Today, I want to consider a related question:

What is the minimum cardinality of a family ${F\subseteq\prod\mathfrak{a}}$ such that ${F}$ is ${<\sigma}$-cofinal in ${\mathfrak{b}}$ for every ${\mathfrak{b}\subseteq\mathfrak{a}}$ of cardinality ${<\theta}$?

Let us agree to denote the above cardinal by ${{\rm cf}^{\sigma}_{<\theta}(\prod\mathfrak{a})}$ (again, don’t blame me for the notation!).

Ok, so why on earth would I want to consider such a cardinal? Well, we’re going to see that this is actually quite relevant for cardinal arithmetic — we will show that such cardinals are intimately tied up with Shelah’s covering numbers. Our first task, though, will be to characterize such cardinals in terms of pcf theory. The characterization will require a little terminology.

Definition 2 Suppose ${\mathfrak{a}}$ is a progressive set of regular cardinals, and let ${\sigma}$ and ${\theta}$ be cardinals such that

$\displaystyle \aleph_0\leq{\rm cf}(\sigma)=\sigma<\theta\leq|\mathfrak{a}|. \ \ \ \ \ (2)$

We define the set ${{\rm pcf}_{\Gamma(\theta,\sigma)}(\mathfrak{a})}$ to consist of all (regular) cardinals ${\kappa}$ for which there is an ideal ${J}$ on ${\mathfrak{a}}$ such that

• ${J}$ is ${\sigma}$-complete,
• ${(\prod\mathfrak{a}, <_J)}$ has true cofinality ${\kappa}$, and
• there is a ${\mathfrak{b}\subseteq\mathfrak{a}}$ such that ${|\mathfrak{b}|<\theta}$ and ${\mathfrak{a}\setminus\mathfrak{b}\in J}$.

Notice that the above is very close to the definition of ${{\rm pcf}_{\sigma{\rm-complete}}(\mathfrak{a})}$; the difference is that the last clause further restricts the acceptable ideals to those which also “reduce” ${\prod\mathfrak{a}}$ to something of the form ${\prod\mathfrak{b}}$ where ${|\mathfrak{b}|<\theta}$.

We’ll end this post with the statement of a theorem; we’ll prove the theorem in the next post.

Theorem 3 Suppose ${\mathfrak{a}}$ is a progressive set of regular cardinals, and

$\displaystyle \aleph_0\leq{\rm cf}(\sigma)=\sigma<\theta\leq|\mathfrak{a}|. \ \ \ \ \ (3)$

Then

$\displaystyle {\rm cf}^\sigma_{<\theta}(\prod\mathfrak{a})=\sup{\rm pcf}_{\Gamma(\theta,\sigma)}(\mathfrak{a}). \ \ \ \ \ (4)$